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VATHEMATIC® Linpsre 


THe design in preparing this treatise on the Differential and In- 
tegral Calculus has been, not so much to produce a work that should 
cover the whole ground of this extensive and rapidly extending branch 
of mathematics, as to produce one that should be complete within the 
limits assigned it, and adapted to the wants of students in the higher 
schools and colleges of this country. Many of the subjects are much 
more fully discussed in this volume than in other elementary trea- 
_tises; while many are entirely omitted here which are generally 
included in such works, though they are not essential to, and are 
rarely embraced in, the college course in this or in other countries. 
The necessity devolved on the author, either to be limited in the num- 
ber and full in the treatment of the subjects selected, or full in the 
number of subjects, and limited in their discussion. The former 
choice was taken, keeping in view the logical and progressive develop- 
ment of the principles. 

This will account for the omission, among other subjects, of the 
integration of differential equations of the different orders, and of 
the ‘* Calculus of Variations,” the latter of which, when fully treated, 


would make a volume equal to the present in size. 


4 PREFACE. 


It will be found, however, that the time usually given to this study 
will render it impossible to take, in course, all the subjects herein 
treated. The following are what may be left out in the class-room 


without serious breaks in continuity : — 


DIFFERENTIAL CaxcuLus. — Part First.— Section V., from Article 
68 to the end of the Section. The whole of Section VII. Section 
X., from Article 110 to the end of the Section. The whole of Sec- 
tion XII. Section XIV., from Article 139 to the end of the Section. 

DIFFERENTIAL CatcuLus. — Part Second. — The whole of Section 
II. Section IV., from Article 177 to Article 181; from Article 188 
to the end of the Section. 

INTEGRAL CaucuLus. — From Example 4, Section IV., to the end 


of the Section. The whole of Sections IX. and X. 


It will be observed that the fundamental proposition of the Differ- 
ential Calculus is based on the doctrine of limits; and that of the 
Integral Calculus, on that of the summation of an infinite series of 
infinitely small terms. The author adopts these methods merely on 
logical grounds, but ventures the opinion that these, and what are 
called the infinitesimal methods, are based on the same metaphysical 


principles. | 
THE AUTHOR. 


NOVEMBER, 1867. 


DIFFERENTIAL CALCULUS==-=—- 


pe ASE Ly 2 FE, 
SECTION I. PAGE. 
GENERAL PRINCIPLES AND DEFINITIONS . . : . : ° Sn a 


SECTION: Tk 
DIFFERENTIAL CO-EFFICIENTS OF EXPLICIT FUNCTIONS OF A SINGLE VARIABLE, 23 


eo. CT LON EL. 


DIFFERENTIAL CO-EFFICIENTS OF INVERSE FUNCTIONS, FUNCTIONS OF FUNC- 


TIONS, AND COMPLEX FUNCTIONS OF A SINGLE VARIABLE . : 2 Z . 41 
SECTION: Ty: 
SUCCESSIVE DIFFERENTIAL CO-EFFICIENTS ., : ; 2 r 7 ‘ ‘ BO 


5.0 TE TO NV. 


RELATIONS BETWEEN REAL FUNCTIONS OF A SINGLE VARIABLE AND THEIR DIF- 
FERENTIAL CO-EFFICIENTS.—TAYLOR’S AND MACLAURIN’S THEOREMS ., - 69 


SECTION - VI. 


EXPANSION OF FUNCTIONS ,. « i 


Be ON Whe 


- APPLICATION OF SOME OF THE PRECEDING SERIES TO TRIGONOMETRICAL AND 
LOGARITHMIC EXPRESSIONS . : i a eeat A ’ e vs : . ae EY, 


S ECcoLOone VIL: 


DIFFERENTIATION OF EXPLICIT FUNCTIONS OF TWO OR MORE INDEPENDENT 
VARIABLES, OF FUNCTIONS OF FUNCTIONS, AND OF IMPLICIT FUNCTIONS OF 
SEVERAL VARIABLES A - - : : - F - 7 J - . BEA lr: 


SECTION IX. 


SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE INDEPENDENT 
VARIABLES, AND OF IMPLICIT FUNCTIONS. ° ° . . ° . - . 133 


SECTION X. 


INVESTIGATION OF THE TRUE VALUE OF EXPRESSIONS WHICH PRESENT THEM- 
SELVES UNDER FORMS OF INDETERMINATION , ° - ' ‘ pe A Paty, 


SECTION XI. 


DETERMINATION OF THE MAXIMA AND MINIMA VALUES OF FUNCTIONS OF ONE 
VARIABLE . . s e s e . . . . s e . . ° e 176 


SECTION XIL 


EXPANSION OF FUNCTIONS OF TWO OR MORE INDEPENDENT VARIABLES, AND 


. ° ° . ’ ‘ « 90 


INVESTIGATION OF THE MAXIMA AND MINIMA OF SUCH FUNCTIONS. ‘é - 188 
ee OLE = ee 
CHANGE OF INDEPENDENT VARIABLES IN DIFFERENTIATION , = d e2it 


PS lk Od fad Se. @ 
ELIMINATION OF CONSTANTS AND ARBITRARY FUNCTIONS BY DIFFERENTIATION, 226 
5 


6 CONTENTS. 


PART Ty 


GEOMETRICAL APPLICATIONS. 


SECTION IL. Lp a 
TANGENTS, NORMALS, SUB-TANGENTS, AND SUB-NORMALS TO PLANE CURVES , 238 ~ 


Sie GLO No 


ASYMPTOTES OF PLANE CURVES.—SINGULAR POINTS. —CONCAYVITY AND CON- 
VEXITY e . . e e e e . es e e e e . . . 249 


SECTION III. 


POLAR CO-ORDINATES. — DIFFERENTIAL CO-EFFICIENTS OF THE ARCS AND AREAS 
OF PLANE CURVES.—OF SOLIDS AND SURFACES OF REVOLUTION , 5 - 266 


SECTION IV. 


DIFFERENT ORDERS OF CONTACT OF PLANE CURVES. — OSCULATORY CURVES.— 
OSCULATORY CIRCLE. — RADIUS OF CURVATURE. — EVOLUTES ANDINVOLUTES, 281 


, 


INTEGRAL CALCULUS. 
SHCTEON SRE 


MEANING OF INTEGRATION. — NOTATION. — DEFINITE AND INDEFINITE INTE- 
GRALS.— DIRECT INTEGRATION OF EXPLICIT FUNCTIONS OF A SINGLE VARIA- 
BLE.— INTEGRATION OF A SUM.— INTEGRATION BY PARTS.—BY SUBSTITUTION, 315 


On Ons tan OB ae 8 


INTEGRATION OF RATIONAL FRACTIONS BY DECOMPOSITION INTO PARTIAL 
FRACTIONS . : 2 ' ° F ° F : ° * . . F “ . 343 


SE.C TION ei 


FORMULZ FOR THE INTEGRATION OF BINOMIAL DIFFERENTIALS BY SUCCESSIVE 
REDUCTION . . ° . . . F ° z . * r A - » 360 


SECTION oly; 


GEOMETRIC SIGNIFICATION AND PROPERTIES OF DEFINITE INTEGRALS. — AN- 
OTHER DEMONSTRATION OF TAYLOR’S THEOREM.— DEFINITE INTEGRALS IN 
WHICH ONE OF THE LIMITS BECOMES INFINITE.— DEFINITE INTEGRALS IN 
WHICH THE FUNCTION UNDER THE SIGN / BECOMES INFINITE. — DEFINITE 
INTEGRALS THAT BECOME INDETERMINATE, — INTEGRATION BY SERIES . O74 


SEC TIGR. 
GEOMETRICAL APPLICATIONS. 
QUADRATURE OF PLANE CURVES REFERRED TO RECTILINEAR CO-ORDINATES. — 


QUADRATURE OF PLANE CURVES REFERRED TO POLAR CO-ORDINATES . - 392 
SECTION. VI. 
RECTIFICATION OF PLANE CURVES . ° ° ° . ‘ . ‘ ° R . 405 
SECTION VIL. 
DOUBLE INTEGRATION.— TRIPLE INTEGRATION . e « . e e F « td 
SECTION © VII. 
QUADRATURE OF CURVED SURFACES. — CUBATURE OF SOLIDS . ‘ : ‘ » 417 


SECTION IX. 


DIFFERENTIATION AND INTEGRATION UNDER THE SIGN “,.— EULERIAN INTE- 
GRALS. — DETERMINATION OF DEFINITE INTEGRALS BY DIFFERENTIATION, 
AND BY INTEGRATION UNDER THE SIGN /. ° . . . . . . . 429 


SECTION X. 


ELLIPTIC FUNCTIONS : F ; x : 4 F ‘ ‘ ; A ° ° » 465 


DIFFERENTIAL CALCULUS. 
LEAS CONE 


1S NGA See Shed ee oi ed Ae 


SECTION I. 
GENERAL PRINCIPLES AND DEFINITIONS. 


1. In the branch of mathematics of which it is now pro- 
posed to treat, we have to deal with two classes of quantities, 
—constants and variables: constants, which undergo no 
change of value in the investigations in which they are 
involved; variables, which may pass through all values 
within limits that may be restricted or indefinite. 

Variables are usually represented by the final letters of the 
Roman alphabet; and constants, by the first letters of this, 
and sometimes, also, of the Greek alphabet. 

2. When variable quantities are so connected, that, one or 
more -of them being given, the values of the others become 
fixed, the latter are said to be functions of the former, which 
are called the independent variables, or simply the variables. 
The functions are also called dependent variables. 

Thus, in the equation 

y — ax’? + bee, 
y is a function of x, and in this case becomes not only fixed, 
7 


8 DIFFERENTIAL CALCULUS. 


but known, so soon as a value is assigned to x. So also, in 


the equation 


y — ax + bx +cz +d, 


y is a function of the two variables # and z, and is known in 
value when values are given to x and z. 

3. An Huplicit Function is one in which the depend- 
ent variable is given directly in terms of those which are re- 
garded as independent. 

In the examples given above, y is an explicit function of # in 
the first, and-of « and z in the second. In general reasoning, 
when we are not concerned with the particular form of the 


function, explicit functions are denoted by the symbols 


y = Lia), y=f(x), y= 9,4), &e. 


4. An Implicit Function is one in which the relation 
between the function and the independent variable or variables 
is expressed by an equation that has not been resolved in 
respect to the function. 

Thus ax + by +ce=—0, 

ax” + bry + cy* + dx + ey +f=0, 
x? —az?+ cyxz +d =), 


are equations which require solution to render the variable, 
taken as dependent, an explicit function of the independent 


variables. Such functions are also designated by the symbols 
ta, y) =O; oe; Ua ae 


5. Functions are also classified, in reference to their com- 
position, into semple or compound, according as they are the 
result of one or of several operations performed on the varia- 
bles. They are algebraic, when, in the construction of the 


GENERAL, PRINCIPLES AND DEFINITIONS. 9 


function, the only operations to which the variables are sub- 
jected are those of addition, subtraction, multiplication, divis- 
ion, involution denoted by constant exponents, and evolution 
denoted by constant indices; transcendental, when, in the 
composition of the function, the variables have been subjected 
to other operations, combined or not with those regarded as 
algebraic. 
mee ey log. ae, yy = Sin. a, 4 = smn! a,* 


* 


are examples of transcendental functions, and are exponential, 
logarithmic, or circular, depending on the mode in which the 
variable enters the functions. 

6. A function may be continuous or discontinuous. It is 
continuous, when, by causing the variable to pass gradually 
from any value to another separated from the first by a finite 
interval through all the: intermediate states of value, the func- 
tion will itself pass gradually through all the values interme- 
diate to those corresponding to the extreme values of the vari- 
able; and when, besides, the law of dependence of the function 
upon the variable does not change abruptly in the interval. 

y —f'(x) is continuous, if, by giving to « the infinitely small 
increment h = aa, y receives the infinitely small increment 
Ay=—F(x+ax)— F(x). When the law of the function is 
such that these conditions are not satisfied, the function is 
discontinuous. 

“7 The Limit of a Function is the value towards 
which it converges, and from which it finally differs by less 
than any assignable value, when the variable upon which it 


depends itself converges towards some fixed value. 


* Read arc whose sine is x, and frequently written arc (sin. = x). The notations 
cos.—' x, tan.—'z, &c., have like significations. 
2 


10 DIFFERENTIAL CALCULUS. 


It is of the highest: importance that we should have a clear 
conception of the nature of limits as above defined, as this 
conception is at the foundation of the differential calculus as 
developed in the following pages. The following examples 
will illustrate the meaning of limit, and give distinct notions 
on the subject to those who have not already formed them : — 


1st, In the geometrical series 


$4+4+4+4éc., 


the sum S of the first m terms is given by the formula 


ga80—r) 40 0) 
1l—r 1—} Fk 

and it is obvious, that, as 2 increases, (4$)” decreases; and, when 
n becomes greater than any assignable quantity, (4)” becomes 
less than any assignable quantity. In the language of the 
definition, as converges towards infinity, S converges 
towards unity. Hence the limit of the sum of this series, 
when v is indefinitely increased, is 1. 

od, The ratio of an are of a circle to its sine has unity for 


its limit when the arc converges to zero; that is, limit 
sin. & | 


1. For it is plain in the first place, that, for sensible 
x 


values of «, the sine is less than the arc. And again: since the 

triangle formed by the radius, the tangent, and the secant, has 

for its measure 4 #. tan. x, while the corresponding sector is 

‘measured by 4 &. x, it follows that the are 2 is less than tan. a. 
Therefore 


sin. x a sin. x sin. & 


x x x tan. x 


sin. & 
Hence we conclude, that, for sensible values of the are a, a 


GENERAL PRINCIPLES AND DEFINITIONS. Teh 


is always included between two ratios, both of which have 
unity for their limit. It must, then, have the same limit; and 


_ -sima 
we have lim. tl I 
a 
Again : 
sin. x 
. e Hb; 
SIn. @ COs. x tan. # 
a a cos. ©, and 
46; x 4h 
Penn 2 ; t ‘ 

lim poaiees LI cos. 3 but lim. cos; a ==. 1, 


} eg tan. 
therefore the limit of sari 


must also be unity; 7.e., the limit 


of the ratio of an arc to its tangent is unity. 

Cor. The limiting ratio of the arc to its sine, and of the arc 
to its tangent, each being unity, it follows, that, when the arc 
is infinitesimal, the arc and its sine, and the arc and its tan- 


gent, may be regarded as equal. 


3d, For another example, let us take y = , and trace 


x 
1l+2 
out the series of values which y assumes when positive values 


are given to x Beginning with x—0,we havey—0. By 


whi 1 Lae 
division, the value of y takes the form 1— ——-; from which it 


ctl 
is seen, that, as x increases, the subtractive part of the value 
of y decreases, and y itself increases ; and, as # approaches 
+ oo, y approaches its limit 1: but, for all finite positive values 
of x, the values of y are less than 1. The difference between 
y and this limit can be made as small as we please by giving 
to x a value sufficiently great. Thus, if we wish to make this 


1 
] ——— = 
difference less than 1,000,000" we make x = 1,000,000 


In this same example, let us now give to x negative values, 
and observe the changes in the value of y as x increases nega- 


12 DIFFERENTIAL CALCULUS. 


tively from 0, and approaches — oo. Replace x by —4#, then 
x ane. y 
fia ei Waa ale TG Beer 
and let us consider the values of y answering to values of ¢ 
between the limits 0 and 1. Beginning with ¢ = 0, we have 
y —O0: for all other values of ¢ between these limits, the 
denominator of y being negative, y is itself negative. As ¢ 
increases, ¥y Increases numerically ; and, when ¢ differs from 
unity by less than any assignable quantity, y is greater 
numerically than any assignable quantity; that is, —o is 
the limit of y for ¢ = 1, which answers to x——1. This is 


equivalent to saying that y has then no finite limit. 


; : t 
When ¢ passes 1, the denominator of the fraction —-— be- 


comes positive, and y changes from negative to positive. In 
this case, y passes abruptly from — o to + o while ¢ is pass- 


ing through the value 1. The value of y may now be put 


ye : 
under the form y = ———. For all finite values of ¢ greater 


i halbees 
t 


than-+-1, y is greater than 1: but y decreases as ¢ increases; 
and finally, when ¢ becomes greater than any assignable quan- 
tity, y will differ from its limit, unity, by less than any assign- 
able quantity. 

Trigonometry furnishes a case of limit similar to that of 
this example, when ¢ passes through the value unity. As an 
arc increases continuously from 0, its tangent also increases 
continuously, but. more rapidly than the arc; and, as the arc 
approaches 90°, the tangent approaches its indefinite limit 
+o. When the arc passes through the value 90°, the 
tangent changes suddenly from an indefinitely great positive 


to an indefinitely great negative quantity. 


GENERAL PRINCIPLES AND DET TOaS. 3 


P 
8. The exact meaning of the word «limit » will be under, 


stood from what precedes; but it is well to tall attention to \, 


abbreviations of expression frequently sed in this/ connéc. ’ 


tion. In finding the limit of" when « Nore 


7 | 
out limit, it would be said, we ae 


POPS N ae 
limit 


= 1 when «= 0; 


but it must be borne in mind that ats 


a } 
cannot reach this 


limit so long as a has any value. And, if we actually make 


Re aine 2 : 
x = 0, the ratio has no meaning ; in fact, ceases to exist. 
Ai 4 


It is true, that if « be not supposed to vanish, but simply to 
differ from 0 by less than any assignable quantity, that is, if x 
becomes infinitesimal, the ratio retains its significance, and its 
value will differ from its limit unity by less than any assign- 
able quantity. 


In this case, the language is an abbreviation for this or its 


equivalent: “As a is diminished, the ratio 


sin. & 
converges 

x 
towards unity, and can be made to differ from it by as small 
a quantity as we please by taking « sufficiently near zero.” 
And, in all similar cases, the language is to be interpreted in 
the same way. 

In other cases of limits, the inconsistency just pointed out 


does not present itself. Any finite value of y, in the example 


“ 
LY tiem ey that answers to an assumed and finite value of a, 
x 


may be taken as a limit of y; and it would be strictly correct 


to say 


a 1 
limit of ——— = — when zw = 1. 


z+ i 2 


This corresponds to the definition of limit given in Art. T. 


y 
4 


y 


14 DIFFERENTIAL CALCULUS. 


9. Rules for the evaluation of functions, which, for particu- 


lar values of the variable, assume the indeterminate forms 
0 : ; 
0” oe 2 ,0 x wo, 0°+ 1%, will be established in a subsequent 
section: but itis necessary for our purposes to consider in this 
1 
*place the function y = (1 + a), and to find its limiting 
value when x = 0; the function then taking the indeterminate 
form 1°. 

The variable « may converge towards its assigned limit 
zero through either positive or negative values. Let us first 


is cee 
suppose x to be positive, and represent it by the fraction —;. 
m 


then, as x diminishes, m increases ; and, when « becomes a very 
small quantity, m becomes a very great quantity. 

If m be an entire positive number, we have, by the Binomial 
Formula, 


(+2) =(145 ny) =1+14 a Go 


m? 


Re gl 
it SP 
m m—-l1 m—2 1 
B ey 3 mit &., 


a development which will contain m + 1 terms. 
Dividing both numerator and denominator of each term by 


the power of m that enters the denominator, we find 


E z| m 1 td 1 9 
(+2) e € + . =245(1 5) +33(1—<) (12) 


ei 1 2 33 
+3537—z) (1—=) (1—S) + og 


Under the hypothesis that m is a positive whole number, the 
2 


33 . o,@ 
expressions 1 — —, 1 — —, 1 — -, &c., will each be positive, 
eat m m 


GENERAL PRINCIPLES AND DEFINITIONS. 15 
oN \n 
and less than unity. Therefore (1 +- =) = 2 + some posi- 


(a>: 


Again: the development will be increased in value both by 


tive quantity; that is 


Le 
neglecting the subtractive terms —, —, m oo? and also by 
m m 


replacing each of the denominators 2, 3, 4, &c., by the least 
denominator 2; that is, the true value of the development is 
less than 2 plus the series 


Digits fia cae 
ORS Ten an as 


But this series cannot exceed 1, however far continued: 
1 


2: 1\” 
therefore (1 + x) 2 (1 +=) is always included between 
m 
the limits 2 and 3. 


n ee 
if — = mis a fractional number, it will be found between 
ae 


two consecutive whole numbers m andn=m-+1. Lets and 
t be two positive proper fractions, whose sum is always equal 


to 1, and make 
LK — 


° m 
m + s =n —t, whence ) 


1 
and the expression (14 ca will be included between 


16 DIFFERENTIAL CALCULUS. 


Now, as 2 decreases indefinitely, m and increase to infin- 
kof Ty? 1 
ity, and the two quantities € +5) and € a =} both con- 


verge to the same limit, which, as was proved above, is includ- 


ed between 2 and 3; while the exponents 1+ =, 1 Bd, 


n 
converge to the limit 1. 


? 


1 
It follows, therefore, that the two expressions € ot =) 
m 


Dyes ee 
€ 4. a1 have the same limit, and that this limit is the same 


1\™ 
as that of € a. =) when m, regarded as a positive whole 


number, is indefinitely increased. 


Finally, if x is negative, and either entire or fractional, make 
1 : 
oe ani : so that, for all values of x numerically greater 
Z 


than 1, z must be negative and included between the limits 0 and 
1; but, for values of x less than 1 (and it is with these alone 
that we are now concerned), z must be positive, and increase 
to infinity as # decreases to zero. Making this substitution 


for x, we have 


eae 1 SAE eae 2+] z+l 
(i+e)*=(-775) Sn 
z+1 1X6 i++ 


Hence, when « approaches its limit zero through negative 
1 


values, the limit of the expression (1 + 2)? ee 


i Aw) ark _ 
€ + =) is the same as when this limit is reached 
Z 


by causing x to decrease through positive values. 


GENERAL PRINCIPLES AND DEFINITIONS. iy. 


To find what this limit is, resume the equation 
(adfase BHO DO-2 
Hasta) 5) 5) + 
eee 


and suppose m to be wee. then 


, 1 Lae I 1 sed a Teer} & 
im. ( $2)F = 45 Hy eR 4 eB ee 
2 — 2:0000000 ... 
fe — ‘5000000... 
2 
1 
es — ‘1666666... 
2:3 
1 
BT et te, — -0416666... 
pay wae 
1 
pvericadl a — -0083333 .. 
9.3.4. 
3 ' 
— 0013888 ... 
2.3.4.5.6 
1 
pee eee ee O01 984). - 
2.3.4.5.6. / eee pa 
, - = 0000248... ... 
2-4.4.5.6.1.8 
1 
se — > Dike: 
9.3.4.5.6.7.8.9 Ne 
Sum = 2°7182818 . .. ,anumber that is incommensurable 


with unity. 
It is the base of the Napierian system of logarithms, and 


will be denoted by e: 
3 


18 DIFFERENTIAL. CALCULUS. 


Therefore 


1* 


(1+ 2)?_, = 27182818 =e. 


The symbol 7 will be hereafter used to designate Napierian 


logarithms, and Z will denote other logarithms. 


1 
10. Taking the Napierian logarithm of (1+ a#)2, we have 


| Bp ie I 
iL @) = Clie eee 
and, passing to the limit a making « = 0, we have 


ax 


1 
x 


Uil+a)jeslea. 
In any other system of logarithms, we should have 


L(-e)f =— Lan) = 249, 


and at the limit 


Dili dae wee cee 1 

lim. Te ee Cate lim. L(t ee)? ee Le = ja? 

a being the base of the system characterized by Z, and ob- 
serving that, since the logarithms of the same number in two 


systems are as the moduli of those systems, we have 


Te: -le 3s: Me), ov ers si ee eee 
Lat late Hes, or ot tas AE 


1 
and therefore Le = MI = ia? the modulus of the system of 
al 


which a is the base. 
11. Since lim. (l + 2)" =e when @ is decreased without 


1 + 
* The notation (1 4 x) x indicates that the value of the expression correspond- 
2= ; 
ing to x =O is taken. 


GENERAL PRINCIPLES AND DEFINITIONS. 19 


1 
limit, we can from this deduce the limit of (1 + az)? in which 


a is any constant quantity. 
Thus, 
Le Ae 
(1+ az)? = («a + az)™ | : 
Now, as 2 diminishes without limit, az will also diminish 
without limit; and therefore 
1 
lim. (1 + az)e =e; 


l 
Zz 


lim. (1 + az) 


— oo 


12. In any system of logarithms, 


1 


e) 1 
Lz) = = LU +2); 


Ll 1 
hy lim. SOT) tim. £0. + 2)* = Lies 
and, if the logarithm be taken in the Napierian System, 
lim. Ae) mnt aed 
z 


13. Resuming the equation 
phe 93 0 Ee 
L(+ 2) = oes 


and making 1 + z= a’, whence (taking logarithms in the sys- 
tem of which a is the base), v= Z(1+ z) and z=a’— 1; 
therefore 
. ) 
Reifel, Pa kets = } 
( = ) a’ — 1 ’ 
or, by taking the reciprocals, 
1 : a°’—1 
Lli+s2) vw 


Now, as z diminishes without limit, so also will v, and they will 
reach the limit zero together ; therefore 


20 DIFFERENTIAL CALCULUS. 


1 Soh, Oa 
——, = Im. ; 
L(+2) v 
1 
putin: ose ae 
L(1i+ 2); Le 
Oh gt —— 1 1 
lim. = — =lawhenv = 0. 
vy Le 


Suppose a = e”, whence m = la; 


and therefore 
lim. cas ee en 
4 , 

14, To define some of: the terms, and explain the meaning 
of some of the symbols, employed in the calculus, let us take 
the explicit function of a single variable ; 

y =f (2), 
and give to x an increment denoted by ax; y will receive the 
corresponding increment 
ay =af(2) =sf(e + a2) —/(2), 
and therefore 
ay _ fle +a) —f(2) 


Aw AX 


. 0 . ° 
When ax = 0, the ratio 47 takes the form o; yet it has in 
AX 


fact a determinate value, which is generally some other func- 
tion of x, and expresses, as will be seen presently, the tangent 
of the angle that a straight line, tangent to the curve of which 
y =f («) 1s the equation, makes with the axis of the variable 
x. This limiting value of the ratio of the increment of the 
variable to the corresponding increment of the function is 
called the differential co-efficient, or derivative of the func- 
tion, and is represented by the notations 


y', f'(2), aa lim, lim, #8 


Aw Ax 


GENERAL PRINCIPLES AND DEFINITIONS. 2% 


It is to be observed that the characters a, d, are not factors, 
but symbols of abbreviation ; the former signifying increment, 
difference, or change in value, without reference to amount ; 
while the latter is restricted to particular increments called 
differentials, having such values, that the ratio of the differ- 
ential of the variable to that of the function is equal to the 
differential co-efficient or derivative of the function. The 


differentials are usually regarded as infinitely small. 
’ Sleep se! ‘ Ane . 
15. Before the ratio == reaches its limit /“(x), it must 


differ from it by some quantity which is a function of aw, and 
which vanishes when Av = 0. We may therefore write, 

Ay x + Ax)—f(a ’ 

eo PEPIN apa) +7 


AX 


and, by clearing of fractions, 


ay = f(x + sa) —f(a) =f’ (aw) aw + pace. 


‘Dg - AY ‘ eats 
From this it is seen, that, as the ratio me approaches its limit 


Jj’ (x), y must approach zero; and when Az, and consequently 
Ay, becomes infinitely small, 7 must also become infinitely 
small, and should therefore be neglected in comparison with 
the finite quantity f’(x~). We shall then have 


Ay dy 
—=/f'(x) = oe Aja (ae) A — dy. 


AL 


It is therefore true, that, when the differential of a function 
is infinitely small, it is sensibly equal to the increment of the 
function. 

These considerations are of importance, and are made by 
many authors the basis of the definition of the differential 


of a function; viz., “The differential of a function of a sin- 


2? DIFFERENTIAL CALCULUS. 


gle variable is the first term in the development of the dif- 
ference between the primitive state of the function and the 
new state which arises from giving to the variable an incre- 
ment called the differential of the variable; the development 
being arranged according to the ascending powers of the in- 
crement.” 

16. The definition of the differential of a function follows 
from that of the differential co-efficient. It is the product of 
the differential of the independent variable by the differential 
co-efficient of the function. 

The object of the differential calculus is to explain the 
modes of passing from all known functions to their differential 
co-efficients, and the application of the properties of such co- 
efficients and the corresponding differentials to the investiga- 
tion of various questions in pure and applied mathematics. 

The operation of deriving from functions their differential 


co-efficients is called differentiation. 


SECTION II. 


DIFFERENTIAL CO-EFFICIENTS OF EXPLICIT FUNCTIONS OF A SINGLE 
VARIABLE. 


17. Iv will be convenient, before proceeding to establish 
rules for finding the differential co-efficients of the different 
kinds of explicit functions of a single variable, to investigate 
certain principles which are applicable to all forms. 

Constants connected with functions by the signs plus or 
minus disappear in the process of differentiation. 

The increments of the function and the variable will be char- 
acterized by the symbol A when they are written in the first 
members of equations ; but the labor of making the transforma- 
tions sometimes required in the second members will be les- 
sened by representing the increment of the variable by the 
single letter h, which will, of course, be equal to az. 

Let y =/(x) +c, and give to the variable in this equation 


the increment A; then 


yt sy=f(e+h)+e; 


therefore Ay=f(«+h)—/f(x), 
Ay  f(@+h)—f(x). 
AD h 


Passing to the limit by making ax =A =—0, 
a AT dy 
lim, —~ ——~ — f/f’ 
wt fie oa de 1); 


‘and dy — f(x ao. 
23 


2A DIFFERENTIAL CALCULUS. 


This differential co-efficient 1s manifestly the same as that 
which would have been found had there been no constant 
united to f(a) by the sign plus or minus. As, from their very 
nature, constants admit of no change of value, c¢ has the same 
value in the new that it had in the primitive state of the func- 
tion, and must therefore disappear in the subtraction by which 
the increment of the function is obtained. 


dy 


rox, aL y=—a+a, reach di) =a 
dy 
AUX. ee a SS eee 


18. lf a function of a variable be multiplied or divided by 
a constant, the differential co-efficient will also be multiplied 
or divided by the constant. 
Let y= (x), 
then ade of (2 +h), 
= of (x +h) — f (x) =c (f(@+h) —f(a)), 
(rie+® —F(2)), 


An 


Passing to the limit, 


a 
lim. Ae iota = 
and dy = of (x) da. 
Again: let 
y ==), 
i . eels 
then y= -(f(@+h)—/(2)), 
ay 1 S(@+%)—s (x); 
Ax h 
hae ee oY 8 =f" (x), 


AL 


DIFFERENTIAL CO-EFFICIENTS. 25 


and dy = f(a) dx. 
d 
Ex. 1 y=actb, =a, dy = ada 
xv ae 1 
Ex. 2. = -— —-=- wet : 
Seren, e Ditas oy ay “a 


19. The differential co-efficient of the algebraic sum of sev: 
eral functions of the same variable is the algebraic sum of the 
differential co-efficients of the separate functions. 


Let y=f(e@)tG(v)ty(w)t..., 
then ytay=/(e+h)+G(eth)ty(eth)+... 
ay =f(@ +h) — f(x) &(9(@ +h) — 9(e)) 
+ (w(@+h) — (a) + eee 
Ste N ie) ret is 9) 


Ax h 


h 
whence, passing to the limits, and using the previous notation, 


= ee 


ee 5 aed SP (Ee) Mirae, 
; Ax L 
and dy —f'(x)dx+ g’(x)dxtw'(x)dr+t... 
= (f(x) £ 9! (@) ew! (2) + ae .) dx 
Ex. 1. y—ax—ber+e 


ix, 2. y = af(xz) + br/— 1 g(a) 


a — af’ (x) + br/— 1 g’ (x) 


dy = (af" (x) + br/— 1g’ (x)) dx. 


26 DIFFERENTIAL CALCULUS., 


20. The differential co-efficient of the product of two 
functions of the same variable is the sum of the products 
obtained by multiplying each function by the differential co- 
efficient of the other. ) 

Let y¥ =S(&) X4(z), 
then y+ay=/(e@ +h) x p(w +h) 

ay =f(% +h) X g(a +h) —f(a) X (a) 
=(S(@ +h) —f(@))9(@ +h) + (9@ +2) — (a) Sm); 
eth xe h) — g(x 
44 fet D—SO) ig 1» 4 eos 
Passing to the ey by? making Aas fies ‘ we see that 


lim. DMT: © ale) = f'(x), lim. p(x +h) =qg(a) 


= o’(x)shenge 


lim: 24 =p : x (2) + 9a) x f(2). 
Dividing this equation by y=/f(a) X g(x), member by 


member, we have 


dy 
dx f'(@) , 9’ (2) 
yf (a), uo (a) 


21. The rule just demonstrated for finding the differential 
co-efficient of the product of two functions of the same varia- 
ble may be extended to the product of any number of 


functions. 
Let y =f(x) X p(@) X w(z), 
and make E(x) =g(x) X w(x); 
then Y = fl DEX OH a), 
and oY — 7" (@) x Fle) + F(a) x fle); 


but ul = F" (x) = q(x) X w(x) + w’ (x) X g(e). 


DIFFERENTIAL. CO-EFFICIENTS. 21 


Substituting, in the value of i for F(x) and F’(x), their 


values, we have 


d 
1p =H (2) w(x) Sf (@) FF (@) v (2) 9! (2) +S (2) 9 (x) v’ (22). 
This process has been carried far enough to discover the law, 
that the differential co-efficient of the product of any num- 
ber of functions of the same variable is the algebraic sum of 
the products found by multiplying the differential co-efficient 
of each function by the product of all the other functions. 
Ex. 1. y=(a+ bx) (¢ — ax) mx 
ee = b(c —ax) mx — a(a+ bx) mz + m(a + bx) (c —azx) 
c m( ac + (2be — 2a*—3abz)ar) 


a= (ac + (2b¢ — 2a? — Sabar)ar) de. 

22. The differential co-efficient of the quotient of two 
functions of the same variable is equal to the divisor multi- 
plied by the differential co-efficient of the dividend minus the 
dividend multiplied by the differential co-efficient of the 
divisor, the result divided by the square of the divisor. 

teeliftx Tia +h) 
biti g@y O88 YT gw FR) 


fle +h) fS@) 


OT gah) g(a) 
Bihet+ h) p(@) —g(@ +h) fe). 
g(a +h) g(a). 
= (fe +h) —A@)) 9(@)—(9(@ +4) —9(@)) Aa) 
g(x +h) p(x) 
therefore 
= Kx Oe fle | (a) — $e) pa) 


p(x+h) g(x) 


28 DIFFERENTIAL CALCULUS.: 


Passing to the limit, by making aw =h=0, 
tim, 2% — a _ f'(2) oe) = 9 @) fle), 


BCs Dy (( yx) 
This result may also be obtained thus: 
=75) N&)=Y 9(%); 
therefore, by Art. 20, 
J (%) = y' 9(%) +  (wYy; 
yf (2). 2 SO) 


therefore y! = ae 
da p(x) (9 a 2) 
= f'(£) p(&) —F (a) 9! (a). 
(9(2)) 
Bx. 1. ye an Le 

dy _(b+ax)b—(a+bx)a_ b6?—a@ 
das (b+ ax) = 6+ ax) 
i) eee 

hie (b+ ax)? 


23. The rules which have been thus far demonstrated in 
this section are independent of the form of the functions char- 
acterized by the symbols f, g, w, &c.; and it has been assumed 
that the differential co-efficients of these component functions 
of the compound functions considered can be obtained in 
all cases. Before showing that this assumption is correct, by 
the actual differentiation of all known forms of simple functions, 


~~ : d. 
it is proper to make a few observations on the symbol oe , used 


da 
to denote the differential co-efficient of the function y of the 


variable a. 


DIFFERENTIAL CO-EFFICIENTS. 29 


In the doctrine of limits, dy represents the limit of the ratio 


di 


eo and it must be borne in mind, that, at this limit, aw, and . 
Ax 


consequently a y, become zero. There would be, therefore, an 
inconsistency in viewing dx and dy as the representatives of the 
terms of a ratio that have vanished, until it be proved that the 
ratio itself does not also vanish. If the ratio remains, although 
its terms disappear, then dx and dy may be taken as indeter- 
minates, having for their ratio the final ratio of the vanishing 
quantities. This is the view to be taken of the differentials dx 
and dy, according to definition, Art. 14, and which justifies us 


in regarding these differentials as the terms of a fraction in < 

24. Analytical geometry furnishes instructive illustrations 
of the meaning of differential co-efficients, as was intimated in 
Art. 14, and suggests many useful applications that can be 


made of the doctrine of limits. 
Whatever may be the nature of the function y = f(a), every 


value of x that will give a real value for y will be the abscissa 
of a point of a curve of which y is the corresponding ordi- 
nate; and, if the assumed value of x gives several real values 
for y, x will be the abscissa of a like number of points of the 
curve, having for their ordi- 
nates the several values of 
y. The curve is, therefore, 
the geometrical representa- 
tive of the relation between 
« and y in the equation y 

In the figure, suppose 
SP’P to be the curve represented by the equation y =f(z), 


30 DIFFERENTIAL CALCULUS. - 


and let PI be a value of y corresponding to the assumed 
value OM for x; then give to @ the increment MM’=h, y 
will receive the increment P’Q = ay, and we have 
 P’Q=ay=f (ath) —f (x) = P/M’ — PM: 
sy _ Se+W—fe)_ PQ 


ADS h Veda 


expresses the trigonometrical tangent of the angle P’PQ, 


PQ 
PQ | 
wuich is the tangent of the angle that the secant line or chord 
PP’ makes with the axis of the variable z. Now, it is evident, 
that, as h = MM diminishes, the point P’ moves along the curve 
towards P, and the secant line approaches coincidence with 
the tangent line 7’7”; and finally, when / vanishes, the coinci- 
dence of the points, and of the secant with the tangent line is 
complete. The tangent line to the curve at the point P is then 
the limiting position of all secant lines which have P for one 


ay +s the lite 


dx 


ing value of the tangents of the angles that such secant lines 


of the points in which they cut the curve, and 


make with the axis of x. 
25. The fraction 47 always represents the ratio of. the 
Ax 


assumed change in the value of the variable to the corre- 
sponding change in the value of the function. These changes, 
when small, are properly called increments ; and it is evident 
that their ratio is the measure of the rate of the increase of 
the function to that of the variable: but it will be seen, that, 
for functions in general, this rate of increase will vary both 
with the initial value of the variable and the value of its 
increment aw. If, therefore, the value of the increment 


Ay 


were left arbitrary, the value of. the fraction = would be 


equally so. But the conception of the limiting value of the 


DIFFERENTIAL CO-EFFICIENTS. Sal, 


ratio removes all uncertainty, and suggests to the mind what 
the rate of change in the value of the function is, in the imme- 
diate vicinity of its value for any assumed value of the varia- 


ble. 
dy 


In the case of the curve, in the last article, the limit + 


dc 


does not depend upon the increment Ax, nor upon the form 
of the curve at finite distances from the point whose co-ordi- 
nates are (x,7), but depends only upon its shape and direction 
within insensible distances from that point. 

26. Let us apply these remarks to the equation y =4/ 2 pat, 
which is the equation of a parabola referred to its axis, and the 
tangent line through the vertex as the co-ordinate axes. Giy- 
ing to & an increment, 

Y TAY =N 2% (x +h) 
AY =+/2p (Va +h — VJ) 


mi /Bp ae )= hp 


ViFhtye) Vethtvye 
(7 TA ea 
Dc Wx thtr/x 
therefore lim. 4.7 — dy A ig Ne em a 


Aa da 2/n V2pe” y 
From analytical geometry, we know that P is the natural 


tangent of the angle that a tangent line to the parabola, at the 
point whose ordinate is y, makes with the axis of the curve. 
27. The differential co-efficient of a function, which is a 
power of the variable denoted by any constant exponent, 1s 
‘the exponent multiplied by the variable with its original 
exponent, less one. “~ 


Bae DIFFERENTIAL CALCULUS. 


Let te ey 
then yt ay=(a@+h)" 


syatetir—araan (14h ah 
Had fae x | U hate 
Ax h | x 


Make “atl 24? and { 1 i: the : 
p= ect y (+3) = a 
Put also (lt) lee ee 


Making these substitutions in the expression for re gat 
becomes 
A ics pete 
Ax t 
Both ¢ and z diminish with hf, and reach the limit zero 
simultaneously with it. Taking the Napierian logarithms of 


both members of the equation (1 + 4)” =1-+ 2 we have 
ni (1+ t)=1(1 +2) 
oe f(1+ z) 
ee ae ey 


tits 
Biba ary... Chere and east both have unity for 


a 


their limit; hence 


qe | 
, aD (1 ae das 
Mel CL Anat a ee l(l+a2)é = maser io) > 
z 
But, since n is a constant, 
4° 2 
lim, 2s, 4p 
nt 
ae as | lim. -= 7 
n 


DIFFERENTIAL CO-EFFICIENTS. 33 


__ dy _ 
ae 
28. The rule of the last article may also be demonstrated 


5 A A Zz 
therefore lim. ~! — lim. #2”! [= ner, 
x 


as follows : — 


x cS, nu 
Let, as before, y = x”, 


then ytay=(x+h)” 
xt h\n 
Ay  (@+h)*—ar _ i x y= 
Poa oe i Nol Sen gaan aaa core aa 


‘Now, whether m be a whole number or a fraction, positive 


or negative, it may be represented by He Z in which p, g,and 
s are positive whole numbers. 
ath 
Make ——— = %.*.h=a2(z — 1), 
x 
A’ A. 
and oe, pe ; 
Ax z—1° 
: A a J as eee 
therefore lim, 2% = lim. @™4 ito 
AX z—1 


Ash converges towards its limit zero, converges towards 


the limit unity, and # and z reach their respective limits simul- 
r-4 


taneously: we have then to find the limit of ar Liga as 


Pane armen ge t 
x P-4 
Make wu =z‘; whencez* = u?4,z=— u'‘,and the limit of u 
is unity also. Making these substitutions, we have 
p-4 
ee uP | a? 148 eee Ue st), 
Gt wt 1 ee t(ut—1)— we t(u?— 1) 
Dividing both numerator and denominator of this last frac- 
tion by wu — 1, it becomes 


iil eee te 1), 
wie buf. FI) 


a 


34 DIFFERENTIAL CALCULUS. 


and, at the limit where w = 1, this reduces to Psy 


LAY 
hence lim. —~- = lim. 7”? —————. = _ =- = —— a” 
AX g— 1 “de 8 


29. The rule proved in Arts. 27 and 28 is general; but, 


when the exponent n is a positive whole number, the demon- 


stration below is more simple than either of those given. 


Let y= a; Xa, K 2.4. @,, 1m which wa ecee ames 


n? 


tions of the variable ~; then, Art. 21, 


* / 
HX Wy XK 2. yy XO, +a X LyX. . eee 
to n terms. 


- / duc ? 
Now suppose “4, = 2 = a@, =. 1) = 2 buenas er teem 
v',y==...== x’; and each term in the value of y’ becomes 
dy 


fs Hence yi = nx, 


thy 
Under this supposition in reference to n, we may develop 


(a +h)” by the Binomial Formula, and thus get an expression 
for the ratio =f of which the limit can be readily obtained. 
Thus y =a", Ay=(x+h)” — x” 


et | 
—nxzth+t+n Ree xe”? h? +. &e. 


AY — ng +n id +; L mh + &c., 


‘ 


Ax a 
in which all the terms in the second member after the first 


term contain i as a factor ; 


hence him: 4 se oe = nav", 


DIFFERENTIAL CO-EFFICIENTS. 35 


"ate 
xl. y=at be’, dg = 30% 
d 2 
Ex. 2. y Se Or, a = — = 
oe ore ae 

Ex. 3. eee oT 

d + 2 

de 2h (x* + a*) — 22° (x?+ a’) 


2x 22° 2a? a 
Fa wa (a tary 
30. The differential co-efficient of a function of the form 
y = a", a being a constant, is the function multiplied’ by the 
Napierian logarithm of the constant. 
Let y = a”, then 
—y fay = att = ata" 
Ay = a7 a* — a* — a* (a* — 1), 


and 


Passing to limit, 


A a h hes’ 
Wee Se 1 im, a pg eae waa L 
AX dic h 
But, by Art. 13, 
' a" oe 1 
lim. —la 


therefore lim, 2 ae 
Ce £OL 
If y =a", then y =(a°)*, 
and ot ht Ta 
any 1: _— ebtten?) - — jlitee”) d (b a ont) 9%exre ( bea”) 
(tL AL 


* The letter d thus written before an expression indicates differentiation. Thus, 


d(atbx? d 
if u=a-+bexr?, then peers) is equivalent to as 
dz dc 


36 DIFFERENTIAL CALCULUS, 


n O. et Roe on 
hier? y= ere ey —er 4 — nee, 
> dx dic 
Ex. 3... y= e*¥=14 ¢-*¥>1 


hs Agar (e*vr1 e*¥=1) 
Fix: 43 Y= ee me — eter. 


31. The differential co-efficient of a function which is the 
logarithm of the variable taken in any system is the modulus 


of the system, divided by the variable. 


Let Tema bo: 
thenytay=L(x+h), Ay=L(et+ h)— La = L2 4", 
Whence Ay L = x : 
Kove ehoah 
Make h = «xz: therefore 
ay _LO +2) _1E0+e 
Ax we x z 


But h and z reach the limit zero simultaneously; and, by Art. 


f mea ee) for 2= 0, is equal 10 Tea 4 


Ho the: limit: 0 
. la 


= M, the modulus of the system: therefore 


iin ee 
Aa” ds x 
; ys wed. 
Hence, if (poy ts ire 
Hx1 meme fi vicltes ay | 6 ee : 
aiie . Y ae, a= FOb ) ie ey 
| : dy 
Ex. 2. Reese == @ + Dalam 


dx 


DIFFERENTIAL CO-EFFICIENTS. 37 


22. The differential co-efficient of the sine of an are is 
equal to the cosine of the arc. 
Let y= sin. x, then 
yt+ay=sin. (x +h) 
Ay =sin. (x +h) — sin. a 


== 2 COB, (2 + 3) sin. f . (Eq. 16, Plane Trig.) 
ey sin. — ; 
TI fi See = 4 
1érerore coe D COs. (2 L ,) ‘ 
2 
But, when / 1s diminished indefinitely, the limit of 
oy) 
sin. — 
2 , hy 
a ==) (Art. .(), and lim. cos, (2+ 3) <= CORO. 
2 
therefore lim. “4 = dy aOR. a. 
Ag 10s 


33. The differential co-efficient of the cosine of an arc is 
equal to minus the sine of the arc. 
Let if =itOs, Wunen 
ytay=cos. (x+h) 
Ay = cos. (a +h) — cos. x 


ane) oa h 
— —¥gin. = sin. & x 
. ( at . 


aed 


Mh ade 
Synth ee 
Ay 2 sn f 
i aa pe sin. (2 a 
3 
pees /t 
sin. - 1 
At the limit ah i Te sit: (2 = be 5) en BIT Be 


2 


38 DIFFERENTIAL CALCULUS. 


therefore 
chee ths, fey dy =o 
lim. A ane 


34. The differential co-efficient of the tangent of an arc is 


=~ Sines 


equal to 1 divided by the square of the cosine of the arc. 
Let y = tan. x, then 
y+ay =tan. (x +h) 
Ay = tan. (%-+h) —tan. « 
sin. (w+ h) sin. & 


cos.(7-+h) cos. x 


sin. (% + h) cos. x — cos. (% + h) sin. « 

cos. (x +h) cos. x be 
~~ sin.(a+th— a) 
cos. (a + h) cos. 


- (Eq. 8 Plane Trig.) 


A sin. h 
~ cos. (a+ h) cos. x 
therefore 
Ay sin.h 1 
Aw . Ah. cosi(ac- fh) coat 
at the limit nee aan nA ae " : 
h cos.(«#-+h)cos.« cos.’ x 
di 1 
hence lite tee - see 


Ax « dae Tecos.6a 
35. The differential co-efficient of the cotangent of an arc is 
equal to minus | divided by the square of the sine of the arc. 
Let y= cot. x, then 
y +ay = cot. (a +h). 
Proceeding with this as in the case of the tangent, we should 


find 


DIFFERENTIAL CO-EFFICIENTS. 39 


26. The differential co-efficient of the secant of an arc. is 
equal to the sine of the arc divided by the square of its cosine. 
Let y = sec. x, then 
y+ Ay =sec. (x +h) 
Ay = sec. (x +h) — sec. x 
io 1 _ 1 ___ cos. x — cos. (w + h) 
— cos.(@-+h) cos.@ cos. @ cos. (a +h) 


ae : sin. (2+ 5) 
a <c0s, @ Cos. ( Eh) 
Therefore 


(Ky. 18 Plane Trig.). 


ig h , h 
i in. = 
Aetna 


Ax h& cos.a cos.(x+h) 
2 


Passing to the limit, 


__ dy sin. 
da co32 x 


lim. 


— tan. @ sec. x. 


37. The differential co-efficient of the cosecant of an arc is 
equal to minus the cosine of the are divided by the square 
of its sine. 

Let ¥ = cosec. x, then y + Ay = cosec. (x +h), anday= 
cosec. (x + h) — cosec. x, and so on; the process being the 
same as in the case of the secant. We should thus find 


dy COS. & 
= : — cosec. x cot. a. 
de sin2a 
38. The differential co-efficient of the versed-sine of an are 
is equal to the sine of the are. 
Let y = vers. « = 1 — cos. a, then we have 
dy _ d.vers. x _d.(l—cos.x)_  _—d.cosse_ 
Sh a Cae dc Br Clot: oo ean 


40 DIFFERENTIAL CALCULUS. 


39. The circular functions whose differential co-efficients 
have been thus far found are called direct circular functions. 
Since the tangent, cotangent, secant, and cosecant may all 
be expressed under a fractional form in terms of the’ sine and 
cosine, their differential co-efficients could have been found 
by the rule of Art. 22. Thus :— 


sin. x 
Ist, y = tan. 2 = —_— 
COs. & 


dy _cos’a-+sin’e¢ | 


dx Cos.2 x cos.2 x 
Cos. & 
2d, y = cot. x = ——-— 
sin. x 
dy sin.’ a + cos.? a 1 
dx sin. x ~~ epee 
1 
3d, y = sec. & = ——— 
COs. x 
dy ssn & é 


ena a tan. 2 sec. &. 
x COS.’ & 


1 
4th, y = coset. = — 
sin. x 
dy COS. & 
“= ———,— = cot. & cosec. &. 
dx sin.? x 


The other forms frequently given to the differential co-effi- 
cients of the direct circular functions will be readily recognized 
by the student familiar with the elementary principles of trig: 


onometry. 


SECTION Ml. 


DIFFERENTIAL CO-EFFICIENTS OF INVERSE FUNCTIONS, FUNCTIONS 
OF FUNCTIONS, AND COMPLEX FUNCTIONS OF A SINGLE VARI- 
ABLE. 

40. THE inverse circular functions are those in which the 
sine, cosine, tangent, &c., are taken as the independent varia- 
bles, the arcs being the functions. They are written y = 
sin. ! 2, y — cos. 1a, y = tan.’ a, &€.; and are read y equal to 
the arc of which a is the sine, cosine, tangent, &c. These 
functions are sometimes written y = arc sin. x, y = arc tan. @, 
een 2180 7 arc (sin. = 7), 7 = arc (tan. = x), &.+ but 
the first notation, being the shortest, and that generally adopted, 
will be uniformly used in what follows. 

41. If we have y= q(«), then the differential co-efficient 
of x, regarded as a function of y, is the reciprocal of the differ- 
ential co-efficient of y regarded as a function of a. 


That is, if y=q(x), then x must be some function of y, 


such as «= w(y); whence x Mg Ge es =p" (4) anid, ac 
cording to the principle enunciated, we should have 

dx : 1 

ee 7),.= pia); 


As an example, take the equation 


y = 9(e) =a" + 2x — 3; 
from which we get 
ot — 2(%+-1). 


XL 


42 DIFFERENTIAL CALCULUS. 


Solving the equation with respect to wz, we have 


e=—ltvW/y+4; 
whence ea ata : ; but+Jf/y1+4 1 
dy Oe ae te 


therefore oe ee te 
dy 2(¢+1)’ 

which accords with the theorem; and we will now prove that 
what holds in this particular case 1s true for all cases. 

Let y=g(ax)... (1) be the given function; and since, 
from the nature of equations, « must also be some function of 
y, Suppose = w(y) . .. (2) to be that function: 

If, in Eq. 1, x receives the increment a, y will receive a 


corresponding increment ay: therefore 


ytay=G(epz)... (3). 

Now, Eqs. 1 and 2 are but different forms of the expression 
of a certain relation between the variables 2 and y; and what- 
ever values of y in Hq. 1 result from an assigned value to : 
x, if one of these values of y be assigned to y in Eg. 2, then, 
among the different resulting values of x, one at least must be 
the value assigned to z in Eq. 1. 

It is therefore proper to assume that # and y have the 
same values in Kiq. 2 that they have respectively in Kq. 1. 
Change, then, in Eq. 2, y into.y+ay, and x into «+ aa, 
these symbols having the same values that they have in Kq. 
3: hence 


a+ax=w(y+aAy)... (4). 
From Eqs. 1 and 3, we have 


Ay g(e + Ax) — g(x) cares: 
Arse Ax , 


DIFFERENTIAL CO-EFFICIENTS. 43 


from Eqs. 2 and 4, 
eee LAY) uy) (6); 


ere eS Aras 
multiplying Eqs. 5 and 6, member by member, then 
AY IGG 06 AX)— ap (’ A — 1) 
per aeee ie tS) Pisa eat) bY) (7), 


Ag Ay? Ax Ay 


. A2 ; 
By the preceding remarks, ad y — —1; and, in the 
LOE = Aas 


second member of Kq. 7, the first factor at the limit becomes 


eo = y’(x); and the second factor, ia —w’(x): hence 
dy dx 
dn dy = 0 (Ye) = 1. 
Wl FL y'(a) =) 
rence accor (er) a dy 
du 


42. If we have 

y=w(z)... (1) 

Be Gk la tye (i): 
then y is a function of x; for, by substituting in the first of 
these equations the expression for z in the second, y becomes 
an explicit function of #. Suppose this to be denoted by 

y=f(%) .- . (3): 
Now, if x, in Eq. 2, receive the increment A a, z will take 
the increment Az; and,in consequence ofthis increment of z, 
. yin Eq. 1 will become y + Ay: hence we should have, from 
Kgs. 1 and 2, | 
ytay=yp(z+az) ... (4) 
ztaAz=g(a1tanr)... (5); 
also, from Hq. 3, 

ytay=f(etaz)... (6). 


44 DIFFERENTIAL CALCULUS. 


From the mode of dependence of the variables, we may 
assume that the symbols a, y, z, Aw, Ay, 42%, have respec- 
tively the same values in all of the preceding equations in 
which they occur. Subtracting Eq. 3 from Eq. 6, member 
from member, and dividing both members of the result by 4 a, 


we have 

AQ ta Aa ‘ 
similarly, from Eqs. 1 and 4, 

Avie Az Beene! 
and, from Eqs. 2 and 5, 

a2 9(e+s2)—9(2) og 

years AL ey eee 


Multiplying Eqs. 8 and 9, member by member, we have, 


because the symbols are supposed to have the same values 


throughout, 
Ay he by we Ae) — (2) ot Ae re 
Asay, tice Az AL 


equating the second members of Eas. 7 and 19 
q s q ) 


Lie + Am) = f(m) «(i ae) (Yo 


AX AZ Ax 


Whence, passing to the limit, 


J’ (%) = w'(z) g(a), 


dy dy dz 
a, dc dz dx 
Hixe dy 2 82 — 58 oo (ee 20 
dy dz > | dy __dydz__ 4 
dey. de”. deer da de 7 


DIFFERENTIAL CO-EFFICIENTS. 45 


By placing for z, in Hq. 1, its value from Eq. 2, we find 
: dy __ 
y = 4x" — 62 — 5, whence a 8x — 6; 


the same result as was found by the first process. 

43. Differential co-efficients of the inverse circular func- 
tions. : 

Ist, Differential co-efficient of y = sin.—'z. 


Since y = sin.-'w, «= sin.y; and therefore, by Art. 41, 


dx pe as 
and therefore, by Art. 41, 

dy epee Lie pry it, ] 

dx COs. ¥ ai W/ 1 — x? 


2d, Differential co-efficient of y = cos.‘ a. 


Here y= cos.-!x gives x =cos.y: therefore, Art. 33, 


dd. : hl Saeco 

de =< _ siny= e/a 
and therefore, by Art. 41, 

dy _ 1 i 


th, 3p SN Gn ie eee 
It would be superfluous to point out the necessity for the | 
sions =, =, before the differential co-efficients in this and 
the preceding case. 
3d, Differential co-efficient of y = tan.“! a. 


From y = tan. ‘a, we have « =tan.y: therefore 


dx . 
fh = EET: sec” y= 1-4 tan.2y (Art. 34) ; 
dy se 1 ; 
and 3 Rea oT a ancaie (Art. 41). 
Whence ame ae 


46 DIFFERENTIAL. CALCULUS. 


4th, Differential co-efficient of y¥ = cot. x. 


From y = cot. a”, we have # = cot. y: therefore 


LE ari Daa ae == = cosec.?.y — — (1) cote Art. 35 
Cy ee har | ( cot.”y) (Art. 35). 
dy Seg at 1 : 


5th, Differential co-efficient of y = sec.—! a. 
From y = sec.—'x, we have x = sec. y: therefore 


da = 50-9 & 207%, Gin 


dy cosy 
a ecOs..o) 1 
and = = a 5 (Art. 4d), 
dx sin.y  sec.2y sin. y 
1 
But sec. y = —-—,, hence cos. y = ie = 1 ; and 
COS. ¥ Set. Wy taae 
tei oa gine Vax? —1 
1—sin?y = 5a sin. y = ps Se aati therefore 


6th, Differential co-efficient of y = cosec.—! zg, 


We shall merely indicate the steps. 


Gl») 4.0 COB I is od ae 
TOO eerey., Cay ean a cosec. y cot. y (Art. 37): 
dy 1 1 1 
——— <= —— » Sm. Y = =; 
dc cosec. y cot. ¥ cosec.y | & 
pa d 1 
e0t.. y= + A/a? — 13-42; fo ee 
xMV x? — 


Tth, Differential co-efficient of y = vers.—! a 
Taking « for the function, we have 


em vers.y = 1 — cosy; 


DIFFERENTIAL CO-EFFICIENTS. 47 


dx OG sa, tk 
; —— 8 
therefore dpe =sin.y (Art. 38), and eT a) (Art. 41): 
ae Bees. : z 
sin. Y . pl cos.” ¥ Ale (1 — vers. y)? 
1 1 
=—t a 
/1—(1— 2) / 20 — a2?’ 
1 
ay >. te 


he hae yc nae 
44, The principle demonstrated in Art. 41 has greatly 
simplified the investigation of the formulas expressing the 
differential co-efficients of the inverse trigonometrical func- 
tions. They may, however, be determined directly, without 
the aid of this principle. 
We will illustrate the manner in which this may be done 


by a single example : — 


Let y= sins) g, then 
ytAy=sin.!(«+h); 
and Ay =sin.~!(« +h) — sin.-'a. 


The second member of this last equation is the difference 
of two arcs whose respective sines are «+ anda; and this 
difference is, by trigonometry (Plane Trig., Eq. 8), equal to 
an are having for its sine the sine of the first arc multiplied 
by the cosine of the second, minus the cosine of the first 
multiplied by the sine of the second. Expressing the cosines 
of these ares in terms of their sines, we have 


Ay =sin.—“' (x +h) — sin. 
= sin ((2+h)/T— 2? — a/ [1 — (a +h)']): 
Ay _ sin.— a ((e@+h)/T=% — © —x¢re/[1— (x +h)? 1) 


e . A Mo h 


48 DIFFERENTIAL CALCULUS. 


“Make g=(ath)/1l—a’—a/[1—(«+h)]: 
Ay __sin.'z sin. —'2 z 


therefor — : 
ees AX h Z h 


Now z and / diminish together, and become zero simultane- 


—l, 


ously. At the limit, a —1. To find what ; becomes at 


the same time, multiply and divide the expression for z by 
(wth) /l— a? + x/ (1 — (x + h)*); then 
ate (7+ hy? (1 — x?) — a? (1 —(x#+ i)?) 


Her) (2 +h)V1— a? 4+ e/[1 — (a+ ny") 
Nek 2x2 +h 
(@ +-A)/1 — ow? +2 v(1 —(e#+ h)*) 


Pass to the limit by making 2 = 0, and we have 


he@Ji_e? Vi—a 
dy paren 
daz /J1 — a 


45. Differential co-efficients of functions of the form y = #° 
in which ¢ and s are both functions of the same variable a. 

Taking the Napierian logarithms of both members of the 

equation y — f*, we have ly=sit. By Art. 42, the differ 

ential co-efficient with respect to x of ly is 

Oly dy. 

au. One 

Do ds a, au ds dene Cae 

ds dx dt. -0iae at inet Oe 


and, by Arts. 20, 42, that of s/é is 
lt 


Now, since the equation ly = slt is true for all values of a, 


DIFFERENTIAL CO-EFFICIENTS. 49 


the differential co-efficients of its two members must also bs 
equal: therefore 


Ldy _ 1,48 s dt. 
yada da ' ¢ dx’ 


dy cise 8 UE 
whence TY (" ee we : 7m) 


OTN ear, Parle 
=t! ee sti = t! (uate ae) 


46. From an examination of the particular cases treated in 
Arts. 19, 20, 45, we deduce this general rule for finding the 
differential co-efficient of any compound function: Differen- 
tiate each component function in succession, treating the others 
as constant, and take the algebraic sum of the results. 

Rules have now been given for the differentiation of all 
known forms of algebraic, logarithmic, exponential,and circular 
functions of asingle variable ; and we have seen, that, in gener- 
al, the differential co-efficients of these functions are themselves 
functions of the same variable. 

47. The following exercises are given that the preceding 
rules may be impressed, and that the students may become 
expert in their application, and familiar with the forms of the 
differential co-efficients of simple and complex functions :— 

1. y= an i — Baa? (Art. 28). 

2. y = abx*® — cx? 

a = 3abx? — 2cx (Arts. 19, 28). 
2 
3. eam ee rs =e 
dy _2ax(b—«x*)? + bax’ (b—-ax’)? 
As (b—«x*)® 
_ 2ax(b + 2x7) 


1 (O ears): ; 


(Arts. 21, 22). 


50 DIFFERENTIAL, CALCULUS. 


4. fetes Ve + 3V/ete= (2+ (w ot oF 
Put 2s=2+(e2-+c)t; theny = 2%, 


dy dy dz 
and ae (Art. 42). 
dy 1 1 
But = = rn] 
dz 1 2 3(a@+c8+1, 
and Se he Bae 6 hee sree 
de Fe 6 BON ON Sees 
therefore oY so fig ee 
Y  2(e+(a+e)*) 3@+e)h 
82/(a cyl 
~ 6 +a be XY (e+e) 
5. ys=l(atvVitea?). Makez =x +WV71+a?: 
then — Vert pa sah? oY = at 
But epee arevias (Artogin 
dz % gti 
Pi ee | Rtg 


dic Has ates 1 oe 
1 oA Tat ee 


therefore dy — 


de e+J/t+a? Vita? Vita 


The utility of substituting a single symbol to represent a 


complicated expression before differentiation is exemplified in 


this and the preceding examples. Oftentimes the labor, both 


mental and mechanical, of the mathematician is greatly abridged 


by the adoption of suitable artifices. 


DIFFERENTIAL CO-EFFICIENTS. 51 


Veet o 
MRL 
nominator by the numerator, then 

y =1(1 + 2a? — 2a /1+2°)- 
Put pee Oe _9¢-4/ 1 + 2? Whence y = lz, 


Multiply both numerator and de- 


Oey b 


dy _dydz, 
ae dx dz dx’ 
dy 1 1 
but es —— 
a dz, --% 1+. I? dar/1 +e? 
em yp spor oa" 
and BSE <tr 2 aR Daa ee 
d 4o VJ 1+2 Vices 
DPS Doe ® a1 ie?) ity 
V1 + x? 
dy 1 2 (1 4- 2a? — Qa / 1-2?) 
ee oa Tt JT a 
oe 2 
~ Via 
3a7a — 2° 
—— =) ° 
{i y= tan. Gat = 8e%) 
3a°x — ae ae %, 
Make gran e then y = tan.—'2, 
dy dydz dy 1 
d ik A 
a dx ~~ dz dx’ dz 1-4-2? 
eat 1 < a'(a? 32) 
Sere 3a? 2 — 732\2 (a? + a3 
| ie acs gaa) 


dz 3(a® — x”) (a? — 3x”) 4+ 6x (3xa? — x) 
dx a(a — 3x?) 
Be (a 2a7e? Fey 3 (a ate 


a (a’— 32° )? a (a? — 3x07)? 


52 DIFFERENTIAL CALCULUS. 


dy _ dy dz _- a? (a* — Ba")? 3 (a? + 27)? 


h = ames aed Bt 1. 8 
therefore Ri at ian (a? (a? pa x a (a? — 3a)? 
; dy _—s-_-33a 
This is as it should be; for 
3a? a2 — a o* 
tan. a (a? — 3a) = Late r : 
x x 
therefore Ulead talenes = Make a then 
d dy d. 
Yrmaotan., 2. and = “ai 
dy 3 3 oie 
but ae — 1 ale a2 = (re se Py, Lo 5 ee (Art. 43), 
] Ete ees dy Bae). 
oe dz: a") da — a eeae 


* To prove this, take Eqs. 28 and 33 Plane Trig., and in 28 make )=2a; 


then 
tan. a + tan. 2a 


1 — tan. a tan.2a ; 


tan. 3a = 


Substituting in the second member of this the value of tan. 2a taken from Eq. 33, 
and reducing, we find 


Cone 8 
tansa== 3 tan. a— tanta 


1—3 tan.2a 


: : ; 8a2x—-x3 
Dividing both numerator and denominator of the fraction 


a% —3ax2 
3 
ot bg 
~ bie Sila Tos 
3 
by a3, it becomes ~ ; and, comparing this with the formula for the 
1— 302 


tangent of 3a, we conclude that it is the expression for the tangent of three times 


the arc of which ~ is the tangent: therefore 


3 
hee hi 
mea oy 8a2x—x3 
x = Paice 
3 tan.—!— =tan.—1 ae a (a? — 3x? ) 
a x2 
1—3 a2 


as was assumed. 


and 


But 


and 


therefore 


da V a2 — 6? Va? — b* cos. & (a + bsin. x)? 


DIFFERENTIAL CO-EFFICIENTS. dd 


e“(asin. © — COS. x) 


4, a?+ 1 


dy 1 


dx a?+1 dic 


e* (asin. « — cos. x) 


d . s 
e“"(a sin. © — cos. x) = ae“ (asin. # — Cos. 2) 
4b é 


+ e%(acos. « + sin. 7) 


—(a?+ 1lje“sin. 2: 


Cie eae 
Fp no Sin 
re 1 . ,b+asin.x 
Maen) me ae bane 
b+ asin. x 1 sei 
a+ bsin.« Siege COO re erate meee Nt 
dy 1 dy dz 
ABM GIs Hide dan 
dy lig 1 
dz eee a) ane 
(Gubase) 


a+ bsin. x 


eae en ? 
/ a? — b? cos. x 


dz__(a+bsin.x)acos. «© —(b + asin. x) 6 cos. x 


(a + bsin. a)? 


__ (@ — 6) cos. & | 


er cy a 


1 a + bsin. x (a? — 6?) cos. # 


1 


" a+t+Odsin. a” 


54 DIFFERENTIAL CALCULUS. 


_, d+ a0cos.% 


Ome Tye ae COS. prey Psa we should find, in 
like manner, 
dy i 


dx atbcos.a’ 
55 tape e* COS. & ; 
a ea i ( + e* sin. x 
Put z=; then y = tan.—!z, 
dy dy dz 1) ee 


and de deala Pas 
dz 


(e* cos.” — e*sin.@)(1-+-e* sin.) — e* cos.x(e*cos.v + e* sin. 2) 
(1 +- e* sin. x) 
__ e” (cos. & — sin. x — e”) 
oa (1+ e*sin. x)? : 


and 14 it ue (1 + e*sin. x)? 
ee +27 1 e*cos.%@ VO (1+ e* sin. x)’ -+ (e*cos.a). 
+( oS. 
1+ e* sin. z) 
cos ATs eain. 2) 
1+ 2e* sin. a +e?” 
dy _e* (cos. @ — sin. x — e”) 
pene dx” 1+ 2e*sin.ate™ ° 
19 _l+e dy  1—2%—2? 
; I= Tag die? oe 
ae dy _, 
13. Yi Ue. Aa 1 + lax. 
LSB eri dy _ . 
14. y= cot: x, =. oy eae 
x dy a? 


(bays a (a — @) dz (a? — 0)" 


16. 
Lj 
18. 


19. 


20. 


21. 


ai 
DIFFERENTIAL CO-EFFICIENTS. 5D 


y =e* (1— 2’), ee e* (1 — 3a*— @?),. 


rs g\ we dy _ a a x 
to) XG) 0 +4) 


“ay ao” dy wo nat 
4~ Ua’ dx (part 
em~—e = d £4 
se y 


~ ef pene a eae 
Acs: —e* 


yar te); 5 eo ev +. @7@ 


y= (4 2)" (b+ 2)", 


SY = (a+0)"—1(b2)4 jm(b+2)+n(a-+a)t. 


22. 


23. 


24. 


25. 


26. 


27. 


28. 


29. 


3 
dpe * —tan.x+a, oY — tan x. 
1 dy @—/] — 2? 
y= ———? er mE a 
et+Vl—a« de /{—e (1 +2071 —a) 
d 
eee ee} 2 RSs ob hg mbes 
y = (a* + x”) tan. Pi dg 2 tan. mist 
2 “x dy nba"! 
= 1] Ua bor) | elon ban) 1a 4 bewy, 
d 1 
=e Tt Se Bs ai ° 
Y Sa aes. dx cos.a 
Bey Ge dy = ax — a . 
W/o + a/ 20’ dx 9N/ ana? (Vat 72)’ 
ete ty 
5 a re dx (l—a2)V1—2? 


se OP yee i) 1. 
di gran? dx (e®—1)? 


56 


30. 


ol. 


32. 


33. 


34, 


35. 


36. 


37. 


38. 


39. 


41. 


DIFFERENTIAL CALCULUS. 


_WitetVvl—e dy Sau 1 : 
Ti eu a as a1 — a 
mes aie dy. a 
L+7Pf—@ : de wV1l—e@? 
1 dy xY 
Wea, Be rae la. 
y= avar—a% dic (a? — aya : 
oa : 
yae(Cr Y <= : . 
L— 2 dix (x +1)? (a —1)3 
pes V1 +a? +71 — 2? dy 2 CS 
“Vita oV/ilog! do oo =) 
y= sins! mo, oY ls 
%. -A—mey 
US Trt Cae 3 
da V/1 — a? 
— plan fn ee Rie. LOT: 1 
ao da ° Le 
wit ; baal _. opt eee 
Of eae ‘* CY e os (Ste x") = *), 
dae e(1 oye 
x —sin.—1a 
Ye, Meanie 
sin. o(1 <—_ ae — 3 (x — sin.—! a) cos. a. 
La aa V1 — @? 
di sin.* & . 
x 
a + btan. "9 dy ab 
Diet| y i a 
Wes htt a” * cos? — D* sin? 5. 
1 1 
nine dy = x=(1 —Ia) 


da uC? 


DIFFERENTIAL CO-EFFICIENTS. 


o7 


i 6°, a — e? oF 
o. dy L 
dy dy x uid 
Cry FE oepapeebs weg Ua 
See a -12, 1 dy 1 
Sei the repay ee, 
J Neer dn tA PS 9e Sie? 
d 1 
<a) 5 lene set, J 
J Via? de W1—2? 
y = tan. /] — x, 
eee ee ee ve 
Ue (G08. /1 2? 2/i—ae DAWES 


48. 
49. 


50. 


54. 


da 
y —tan.—'(ntan. 2), Be 


n 
cos2a + n? sin2Z a 


8 


d x 
aa tan. Wil ge th |= 
Be Pye, Beas ft 
a= tan. ! ae Bue NO aka —. 
lta’ dx 1+462?+ 2! 
2x dy 2 
ak a4 Ca ioe dea Ih 
Pes eat dz Ita" 
ee OU ey ee 
y — sin. L/sin. x, dx Se Malia con0s, wv. 
<p ale 
jane b+ cx” 
ay a(b — cx’) 
(b+ 62”) 4/5? + (2be — a?) 2? + ©? a 
dl 30 te 


58 DIFFERENTIAL CALCULUS. 


. —ly d . er. 
55. y= «w sin. a VA mE sy a One sin 9 63 


Vi—a@ 1 de (l= at 
tan. b dy a” tan. b i 
56. —= sy oommeabri se = eee 
“ Vea? de a2?— x? (a? — x sec.2b)2 
57. Fie PSE dy _ - xb? — a? 
bY — a dar (b? — «?) Va? — @? 
So.) — tals V1 — cos. x oy a a 
/1 + cos. ae da 2 


(a — b*)2 sin. *) dy ~»p (Gaon 


59. beh i) ee 
aie: (ee Ope dx a+ bcos.a 


60. y = sect dy en 
20? — 1 dix /1 — x? 
Pah ee. A RE ae.) ly ee 
Gane haide-pald. SRD ae da’ [ie 
Bo Fensaae 4} Stan ee 4/2 
1— @/2+ 12? de” Pat 
1, (¢+1)? 1 _,2é—1.. F 
63. =e ante wae bowl 
ad aaa RE V3 , in which 
Perea) dy si) heal 


> dx at(1 +a)’ 


64. From the ee 


nmelowmn 


sin. ne 
; : 2 2 
sin.e+ sin. 2~7-+ ... + sin. nx = ——_______, 


prove, by differentiating both members, that 
cos. 2 + 2 cos. 2x te 3cos.3%-+ ... +ncos.naxis equal to 


\ 2 
cae aot te 5 (ain 2a n+1 2) 


sin. — s, sin. 


2 2 


sin? x 
yy} 


DIFFERENTIAL CO-EFFICIENTS. 59 
65. Admitting* that 


sin. sin (2 4-2)in.(“* +2) are sin,("=2n +a) 


sin. mx 
a 9 pena mts 


in which m is a positive whole number, prove that 


cot. x -+ cot.{— +a)+.... cot. m-+- x |—=mcot.me. 
m mm 
* As the equations assumed in this and the preceding example are not usually 
given in treatises on elementary trigonometry, they will be demonstrated in the 
Key to this work. 


SECTION IV. 


SUCCESSIVE DIFFERENTIAL CO-EFFICIENTS. 


48, Tue differential co-efficient of a function, f(x), of 
a single variable, being in general another function, /’(2), 
of the same variable, we may subject this new function 
to the rules by which /’(x) was derived from f(a), and 
thus obtain the second derivative, or differential co-efficient, 
of the original function. The second differential co-efficient 
will, in turn, give rise to a third, and so on; and we thus 
arrive at the successive derivatives, or differential co-efficients, 
of a function. 

The notation by which these successive differential co-effi- 
cients are indicated will be best explained by an example: — ° 


Let us take y =a"; then 


= ty hee a Pe TS es 1st diff. co-efficient. 
d*y WW —2 d dj 1 
—~=y"”a=n(n—1)a** . . 2° diff. co-efficient. 
da? 
any sof OP ans eee ee m diff. co-efficient. 
da™ 
These are the first, second, . . . m'” differential co-efficients 


of the functiony—=/(x). It is sometimes convenient to de- 

note these by writing the function itself with as many dashes 

as there have been differentiations performed: thus /’(a), 

S(x),. .. f(x), are. the first, second, . . . mm ™oditeren 
60 


DIFFERENTIAL CO-EFFICIENTS. 61 


tial co-efficients of f(x), and have the same signification as 
DP osc YO. 

In the example just given, it is evident, that, if m be a 
positive integer, the m™ differential co-efficient will be inde- 
pendent of x, that is, a constant, when m=; and that the 
function will not have a differential co-efficient of a higher 
order than the nm“. In other cases and forms of function, 
there will be no limit to the number of differentiations that 


may be performed. 


d*y d*®y dy 
The symbols Fictia ad cued ee 
RR Cae ae 
ivalent t siseey ae ies and 
equivalent to Py seer tl 


are read second, third, ... m‘" differential co-efficient of y 


regarded as a function of «; and are to be viewed as wholes, 


and not as fractions, having d’y, d*y,. ..d™y, for their 
numerator, and dx’, dx’, ... dx”, for their denominators: 
nor must the indices 2, 3, . . . m, be considered as exponents 


of powers, but as denoting the number of times the function 
has been differentiated. 
49. Successive differential co-efficients of the product of 
two functions of the same variable. Leibnitz’ Theorem. 
Take w= yz, in which y and z are functions of x; then, by 


Art. 20, we have 
du 


dz d 
eC eae 
and, differentiating both members of this equation with respect 
to x, we have 
au dz, dydz dy dz d*y 
det 9 det" de de * de de * de® 
ae dy dz d*y 
=U eit ae det de® 


62 DIFFERENTIAL CALCULUS. 


In like manner, we should find 


du d*% dy d?z dz d*7 d*y 
dist = 4 gs T° ae deat? de dar © * dina” 
and 
d*u d*z dy d*z d’y dz dz d*y d‘y 
dct = Y dat * de dat tet da? 1 * de da Fat 
This has been carried far enough to enable us to discover 
inferentially the laws which govern the numerical co-efficients, 
and the indices of differentiation in the expressions for 
Mu du dtu 
dx?’ dx*’ dxt 


co-efficients and exponents in the Binomial Formula; for, in 


These laws are the same as those for the 


respect to differentiation, y may be regarded as y®, and 
mas 2), | 
To prove these laws to be general, let us assume them 


to hold when » is the index of differentiation. Then 


du = dz dy dvtas. m—1 dyd™ ts 


da 4 gt de damit” 9” de® da™ 
ne) (n= 2) 00. (n= 7-21) cae 


ar 2.3; seks eds das’ dar’ 
(n—1)(n—2)... (n—r) dttly dr—@tbgz 
fs 9.3... r(r-+1) datt! de»—@FD 
d"y 
tee. lene 


Differentiating both members of this equation with respect 


to x, reducing, and arranging the result, we find 


d™+ly ang dy d”x 
dgett  Y ggrat "Toe due Nee 


Qin s(n pion) dehy dre | dat 
> DR (r+1) dart} dum * eae dantl 


DIFFERENTIAL CO-EFFICIENTS. 63 


Now, the laws of the co-efficients and indices in this devel- 


opment are the same as those assumed: to be true in that 
from which it was immediately derived; but, by actual opera- 
tion, we know them to hold when n = 4: they therefore hold 
when n= 5; and so on: hence they are universal. 


As an example of the above, take wu = e“*y ; then, observing 


Npax 


ae ae. We find 


a" u Re Or: n—1 son n—?2 
a é (ary + nar $ a 4+ a ey lees as) 


Now, by examining the expression within the i se 


that 


we discover, that if (« 1. eS y be developed by the Binomial 


Theorem, treating the symbol if as a quantity, and (s.)¥ 
i 


da 
d\2 
(te) 
’ we get that factor of the development of fa bence 


OP ee Oil 6S Y ye in an. 
dat dae (a+a) 9 


is a convenient and abridged form of writing the n™ differen- 


dy wy d"y 


d 
(Cz =) be then replaced by. op er are hae oe 


d 


tial co-efficient of the function wu = e*"y. 


50. If n be a positive whole number, we may prove that 


d"u sd” uv aia ( a) ta ae ( me 


"de® ds® — da®1\" dar 1.2 dae2\" dai) 
d"v 
— do. do... +(—1rus"... (1). 


For let y = uv, in which both w and v are functions of « ; 
then, differentiating with respect to x, we have 

6 9) alam) hee 71 amas 

da da" da +” da’ 


64 DIFFERENTIAL CALCULUS. 


yee _ au dv, 
dx da dx’ 


and the theorem holds when n = 1. 


whence 


Now, differentiate both members of Eq. 1; then 
Cot naeaty | alae a” ( 


Uda?!) Ge dx®. ane) 7 aged ae 
n(n —1)d"—} d?z 
a yy aC a , 
fs dv 
Se Rare TA ae) ie: 


If the theorem holds for y = wv when the index n has some 


assigned value, it will also hold for x ae when 7 has the same 


dx 


value. Changing, in Kq. 1, v into ad we have 


dv d™u _ d” CS dv\ 9 d?v 
dx dx” Clare da) "dat" da? 


Se at Oe = ( i) 


1.2. de—?\" Ge 
d®tly 


— &e.+...+(—1)"u aaagh AY 
Subtracting Eq. 8 from 2, member from member, and redu- 


cing, we find 
d’tiy d®tlay d” 
Lie OE mea CLS are 1) Toa) 
= 9 
+ (n+ 1) — a(u To) ee 


thy 
mre ces 1)"4)u antl” 


Hence, if the theorem is true i any assigned value of 2, 


it is true when the index isn+1. It is true when n=1; it 


DIFFERENTIAL CO-EFFICIENTS. 65 


is therefore true when n = 2; and so on; that is, it is univer- 
sally true. 


EXAMPLES. 
ia dy 1 al) 1 
ee! eo dat a 
Orem ye 1.2.8 
da? a? det a 
dry aa ae eae Sama 
ae 


2. = An. 2, = = OOR) d= SiN. (2 + 5) 
a 


nm 
Scere 
are oe = COS. ch Ray ire ( +>} 


Yy | , 1 
3. Y= C0s.2, 7 = — sin. & = cos. & Ts }; 


/ — e ; = 
n P) 


4. if == COS. 22, — Oe COR. (aa + ) 


5. 4 = tan.¢-+ sec. @, 


dy 1 sin.x 1-+sin.& 1 
dz cosa  cos2x%  cos?a 1—sin.2’ 
d*y COS. & 


da? (1 —sin. x)? 
9 


66 DIFFERENTIAL CALCULUS. | 


3 sin. « — sin. 32% 


el OP ase Ko n ; 
e 
dl” aN n 
—4 = jain (« + I eee sin, (30 +3) 
es 
a Yai 
eta i ee 
DEAT ibe 
8 Tract oll a8 he a | 
d” 1.2.3...(n—1 
9 Yi ger} in; “is —- - ) 
x ay 4a? 
= 2 2 ard gta SP EN ast Bete 
AHO) fiom 6 fae ehea a da! (ae 
SEE ere dty —x ah 
y= Or ORL, dnt = — 4 cos. x2; hence 
di 
ait fy = 
_1l-«z d”y no Le 
Loy <i OM ere 
CY is aENE tg —2)n—2 
—_“<“ = n?(e* + e-*)" — 4n (n — 1) (e? + e777)". 
On aay 
2 
14. y? = 6ec. 2a, pune — By, 
_ ax+b 
UR Reale gatas bd * 
ity renee b+ Ge eae 
dx" 20 \(@—c)*t! (x9 + c)tH/ 
1 
16 Oe i a) 


dy & iy 2: as ae pe ae 


rail 


DIFFERENTIAL CO-EFFICIENTS. 67 


ete S10, 2, 


aril &: kek Lee 7 
a ond EY a yin bein (2+5) 


fare tS 


Mat (ee Qe a 3 
Seca latsin (ot) + heb 


18, Y — tan.) 7 oad fyi ela 
a a da a 


dx?” a Z 
d*y BY mt y 
dat — eae a 


and = 


eri 2. D.. . (7 — cos,( 7 + nse 1)5 Joos. 


da” qa"—} 
Because  tan.-?~ — a — tan.—} * make tan.—!~ — 4; then 
he a x x : 
tan.—! Ses Sa hea, 
eae) 2 
ny nm Tay) 5 i 
{| — — l)s)= id ee cet et Meech eG — no 
cos € +(n ) sin & +- ) sin. (na — n0) 
! a 
=(—1)""'sin. nf: also cos. 2 = ‘< 
a (a? + x)? 


a” 
Y 
cos.” — = 


2 (a? + a) 


Substituting these values in the expression for ce we have 
ape 


Sel Se Ly Lee ate Pee 1) 
(a? +o'y 


sin. nf. - 


da” 


68 DIFFERENTIAL CALCULUS. 


19. Prove that 
ee 1 :)= (— 1)" 1.2.3... n sin. (noe 
ag? a (a? 4 ety? a. 


Proceed as follows : — 


d tan.— ~ ‘ gs 1 1 dt tan 
imate ae ape) 
and, from this point, the process is similar to that pursued in 
the preceding example. 
20. Prove that 
ad” x liad Co pale ne (n+1)0 
da” be = me (ah x) 3 a 
By Art. 49, 


ae x d” 1 Gace 1 
dx” \a? + x] ~ ” clic” a? +e 1% Ggaat 1\ a? + a? 


and finding the value of each term in second member, as in 


last example, we get 


ae x =e (1.2. 3.. .msin. (n +'1)0 


hc a(a? + 2) r 
i Gale. 14. 2.3.0. singe 
a(a? + 2) 
— (= 11.2.8. . aft (m+ 1)0 
(a? + a) a 
21. When y = sin. (msin.~' x), prove that 


d*y dy 
— 7?) _ 4% —a-4% — my. 
(1 0") 2 shee my 


22. When y = acos. lx + bsin. lz, prove that 
d*y dy 
2 — — e 
oe? ~*~ + 2 ieee, eo 
also that - 


poets 7? 
Ft Gat 1a S Pe ie 


SECTION YV. 


RELATIONS EXISTING BETWEEN REAL FUNCTIONS OF A SINGLE 
VARIABLE AND THEIR DIFFERENTIAL CO-EFFICIENTS.— TAY- 
LORS THEOREM.— MACLAURIN’S THEOREM. 


51. Ir the increment ax of the variable x in the function 


y — F(x) produces the increment ay of y, then the ratio 
2, having for its limit fu = Ff” (x), will, before reaching this 


limit, and when Az is very small, take the sign of #”(x); that 
is, it will be positive if the differential co-efficient is positive, 
and negative if the differential co-efficient is negative. But, 
when a ratio is positive, its terms have the same sign; and, 
when negative, they have opposite signs. Hence, if the dif- 
ferential co-efficient of a function be positive, the function will 
increase or decrease according as the variable increases or de- 
creases; but, if the differential co-efficient be negative, the func- 
tion will decrease as the variable increases, and the opposite. 
52. Suppose that the function y = F(x) is continuous be- 
tween the limits answering to the assigned values «= a, 
% = 4%, of the variable, and that the variable passes by insen- 
sible degrees from the first to the second of these values ; 
then, by the foregoing article, the function cannot change 
from an increasing to a decreasing, or from a decreasing to an 
increasing function, unless the differential co-efficient changes 
its sign from positive to negative, or from negative to positive. 


But a function can change its sign only when it passes 
69 


70 | DIFFERENTIAL CALCULUS. 


through zero or infinity. If continuous, the change of sign 
will occur in passing through zero; if discontinuous, in pass- 
ing through infinity. 

53, If the function y= #(x) vanishes for «=a, and is 
continuous for values of « which are indefinitely near a, 
then i | | 

E(@y + Ax) = AGL’ (G5) + Avr... Aeon 
where 7 is a very small quantity when Aw is very small. The 
sign of the second member will therefore be determined by 
that of #”(x): hence, if =a, + Ax differs very little from xy, 
E(x, + Ax) = F (x) >> 0 if ieee 
F(a, + Ax) = F(x) <0 if F’ (x) <0. 

54. Suppose that the two functions F(x), f(x), are real, 
and that they, as well as their differential co-efficients, are 
continuous between the limits, answering to the values a, and 
x, +h of the variable; suppose also, that, between these limits, 
J’ (z) does not undergo a change of sign; that is, for the in- 
termediate values of a, f(x) must constantly be either an | 
increasing or a decreasing function: then the ratio of the 
differences 

B(a,-h) — F(a,), f (ei +h) -—f(*1), 

will be equal to that of the derivatives I’ (x), 7’ (x), when in 
these x has some value between a, and 2, +h; that is, if 0, 
be a proper fraction, we shall have 

E'(¢%,;th)—F(a,)  F’ (#,+6,h) 

J (#1 +h) —f (#1) (2, + Oh) 
To prove this, let 4 be the least and B the greatest algebraic 
F(a) 
J (2) 


tween x,and x, +h; then the two differences, 


values that the fraction can have for values of x be- 


¢ 
DIFFERENTIAL CO-EFFICIENTS. val 


must have opposite’signs for any of these values of 2; and the 
same will be true for 


En" (x) — Af’ (x), FY (x) — BS’ (a), 
because, by hypothesis, 7’ (a) has an invariable sign between 
its limiting values. But these last expressions are the differ- 
ential co-efficients of the two functions 


E(x) — Af (x), # (x) — BY (a). 
One of these functions, therefore (Art. 52), must be constantly 
increasing, and the other constantly decreasing, while the 
values of x are limited by x, and x, + h. | | 
If, then, the value answering to x, be subtracted from that 
answering tox,-+h for the one and the other, we have the 


two expressions, 


F(a, +h)— F(a) —A(f (+h) —F(m)), 
F(a, +h) — F(a) — B(f(@ +h) — f(x); 


one of which must be positive, and the other negative. 
Wherefore it follows, that, if both be divided by f(x, +A) 
— f(x), the quotients 

F(ey+h)—F (am) _ 

J (41+ 4) —f(#1) 

Bia +h) — # aA 

S(t +h) —f(#1) 

me ite ey Leh 4) 

that's, F(a, +h) = Fa) 


than A, and less than B, and is therefore comprised between 


have opposite signs ; is greater 


these greatest and least values of sae ae a But F” (x) and /’ (a) 


being continuous, while x passes by insensible gradations from 


x, to x, +h, the ratio ee) must pass through all values in- 


J’ (2) 


> 


72 DIFFERENTIAL CALCULUS. 


termediate to its greatest and least values. Hence there must 
be some value of «x between v,anda,+ Ah that will render 
the ratio of the differential co-efficients equal to the ratio of 
the differences of the functions. id 
Let 6, be a variable proper fraction: then, from what pre- 
cedes, a value may be assigned it, that, agreeably to our enun- — 
ciation, will cause it to satisfy the equation 
E(x, +h) — #1) _ E(x, + O,h) 
S(tith)—f(m) f(t + he 
565. It has been assumed in what precedes that /’ (a) re- 


tains the same sign between the initial and final values of x ; 


but the proposition is true when the assumption is made with 
reference to £’ (x), instead of f’(x). For, if #” (x) does not 
change its sign, by the same course of reasoning we can prove 


that 
S(a +h) —S(%1) ff (1+ 4) h) 
F(@,--h) — F(a) F (@- 0h): 


whence 
F(a,-+h) — F(a) F’(@,+0:h) 
F(@:+h)—f(a) ~ fi (a+ Oh) 
56. From the theorem established in Art. 54, we deduce 
the following consequences :— 
Ist, If F(x) and f(x) both become zero for the particular 
value « = x,, then 
F(a, +h) F(a, +04) 
Fler+h) F(a Fahy 
2d, If the differential co-efficients up to the (n — 1) ™ order 
of both F(x) and f(x) vanish for « = a, those of the second 
being constantly positive or constantly negative between the 


limits corresponding to x—=a,,«—wa,+h, while the fune- 


tions themselves do not vanish for this particular value of 2, 


DIFFERENTIAL CO-EFFICIENTS. ia 


then, from what has just been proved, we shall have the fol- 
lowing relations : — 
FE’ (a, + Oh) we EM” (x; +O, h) 
af (+ 9, 4) JU (@1 + 92h) 


FY (a+ 0gh) F(a -+ 05h) 
fl (@1 + O,h) fl (a1 + Gh) 


Fe-y (xy + Gey h) tis Fe (x, + 0, h) : 
foamy (xy ar SS h) Re 7? (ay +6, h) $ 


therefore 


F(e; +h) — F(a) F (a, +0,h) 
Feb) —f (a) ~ F (e+ Oak) 


Since, in the reasoning, no condition has been imposed on 6,, 


except that it be a proper fraction, we may omit the subscript 
mn, and thus have 

(ae, +h) —F(a2,)_ &™ (w+ 6h) (2) 

(ti +h) —f(%1) fh? (#1 + Oh) 


If the functions reduce to zero, with their derivatives for 


= @,, we have 


F(x,+h)  F™ (a, + oh) (c) 
Fath) Ff (a+o) 
Making the further supposition that 7, = 0, then 
Lh) « F@ (6h), 
F(h) — f(Gh) 
but, because this is true for any value of h, « may be written 
for h; and thus 


Pi (at) (002) (d) 
J (a) f (G2) # 
38d, The conditions relative to f(x) that have been im- 
10 


74 - DIFFERENTIAL CALCULUS. 


posed in the preceding propositions are satisfied when f(a) 
= (x —«a,)”": whence 

J" (%) = 1 (@ — 2)? 

J (x2) = n(n —1) (x — a)? 


fOr? (2) = n(n— 1)... 2(e — ay 
FOE) 1.2380, nee eee 
Here f(x) and its successive derivatives, up to the (n —1)", 
vanish for « = «,; and since, in Kq. b, the denominator of the 
first member, 
SJ (#1 +h) —f(%1) = (41 — & A)” — (@, — @))" =H", 


we have 


Fh) —Fe@)y= oe 


2.3. en FS 
When n = 1, this gives 
F(x2,+h) — F(x,) =hF’ (x, + Oh). 
If F'(x,) =0 as well as f(x,) = 0, then 


he 
Ft) > o55 


Making 2,—0, and then writing « for in the preceding 


EL (x2; + Oh). 


equations, they become 


E(x) — #(0) = aaa (62) 
F(x) — F' (0) = xf” (6x) 
F(2)=75 en - F (622). 
57, The equation B(x) _ Pe 


F(a) — f(a) (Kq. d, Art. 56), ex- 


pressed in words, enunciates the following theorem; viz.: If 
there be two functions, /’(x), f(x), which, with their differen- 
tial co-efficients, are continuous, and which, with these differ- 


DIFFERENTIAL CO-EFFICIENTS. ex 


ential co-efficients up to the (n — 1)* order inclusively, vanish 
for «= 0; and if, further, the first » differential co-efficients 
of one of these functions are constantly of the same sign for 
values of the variable between zero and another assigned 
value; then the ratio of the functions will be equal to that of 
the n™ differential co-efficients, when, in the latter, some inter- 
mediate value is given to the variable. 

The importance of this theorem warrants us in giving it an 
independent demonstration. 

Let F(x) and f(x) be two functions which vanish for « = 0; 
and suppose, first, that the differential co-efficient f’ (a) of the 
second does not vanish for this value of the variable, and that 
it retains constantly the same sign between «= 0 and «=A, 
which requires that f(x) be continually increasing, or continu- 
ally decreasing, between these limits (Art. 51), and therefore 
constantly positive or constantly negative, since f(x)=0 
when x = 0; and let 4 denote the least and B the greatest of 
Bee) for values of «a be- 
J (&) 
tween zero and f: then the two quantities, gas A 


fe) he 
F’ (a) 


ean B, will have opposite signs; and, since /’ (2) does not 


change sign, the same will be true of the differences, 


E(x) —Af’ (a), #° (%) — Bf’ (a): 


the values assumed by the ratio 


but these last are the differential co-efficients of the two 
functions, 

i'(x)— Af (x), F(x) — Bf(a), 
one of which (Art. 51) must therefore be constantly in- 
creasing, and the other constantly decreasing; that is, since 
‘both F(a) and f(x) vanish for « = 0,one must be constantly 


76 DIFFERENTIAL CALCULUS. 


positive and the other constantly negative between the limits 
answering tox=0,x=h. Therefore, because f(z) is of in- 

variable sign, 
F(x) —Af(#)_ F(a), Fle) — Bf@) _ F(a) 
J (2) Sha) Sas nes J (2) 


are of opposite signs: whence it follows that the ratio of the 


— B, 


functions 1s comprised between the least and the greatest 
values of the ratio of the differential co-efficients. But, if the 


variable be made to pass by insensible degrees from 0 to A, 


Ga! which is by hypothesis continuous, must pass 
tT (x) bi yP } Pp 


through all values intermediate to 4 and B. If then 6 denote 


the ratio 


a proper fraction, it will admit of a value such that the equation 
Hide). Et! (0) 
T(") ff! (92) 


If the differential co-efficients. of both functions, from the 


will be satisfied. 


1st to the (n—1)™ orders inclusively, vanish for «= 0, by 


reasoning upon them as we have upon the functions, we have 


F020) OE ORG ea ay _ ONG, 
fi(0,") f" (09%) f"(032) 9 a 
whence 


TFC an ORT Oe} 
f(x) f (6x) 
58. Itis to be observed that the only conditions upon which 
the equations 


F(x, +h) — F(a)=hF" (x, + 6h), 
F'(x)— F'(0) = ak" (6x), 


F(@) = py5— F (6), 


aero. 
depend, are, that 2’ (x), and its differential co-efficients up to the 
order involved in the equations, should be continuous between 


the assigned limits of the variable. 


DIFFERENTIAL CO-EFFICIENTS. ba 


59. From the equation (2, +h) —F(x,) =AF’(x,+ 6h) 
of Art. 56, it may be shown, that, if the differential co-effi- 
cient with respect to x of any expression is zero for all values 
of x, such expression is independent of «: for, if I’ (a) is zero 
for all values of x, the above equation becomes 

F(a,+h)— F(x,;)=0; or, F(x#, +h) = F(x); 

that is, the function does not vary with, and is therefore in- 
dependent of, x. It is plain, that, if the differential co-efficient 
is not equal to zero, the expression will vary with x Hence 
those expressions only are independent of a variable for which 
the differential co-efficients with respect to that variable are 
zero for all values of the variable. And further: if two func- 
tions have the same differential co-efficient with respect to any 
variable, such functions can differ only by a constant; for 
the differential co-efficient of the function which is the differ- 
ence of these functions is zero by hypothesis: therefore, by 
what precedes, this difference must be independent of x; that 
is, constant. 

60. Suppose F(x) to be real and continuous; then, by 
means of the foregoing principles, we may find the develop- 
ment of this function arranged according to the ascending 
positive powers of &. 

For we have, Art. 56, 

F(x) — F(0) = ak” (6x) = «F’ (0) + fax 
by making F! (62) = F’ (0) +f; 
whence 
F(x) — F (0) —aF’ (0) = Ra: 
from which it is seen that #,x is a quantity that reduces to 
zero when # is zero; and the same is true of F(a) — F’ (0), 
which is its first derivative with respect to x. Its second deriv- 


78 DIFFERENTIAL CALCULUS. 


ative is /'” (x). Wherefore, by the article already referred to, 
on? 
F(ec)—F(0)—aF’(0)=Rye= 3 F'” (0x). 
Making 


F"(60) = F"(0) + Ry, 


then, as before, 


oe? oe? 
2 
and it is evident that 2, a is a quantity, which, with its first 


and second derivatives, 
E(x) — °° (0) — el" (0), 2” (x) — 2” (0), 
vanishes with «, and that its third derivative is #'’”(a): there- 


fore we have 
of a” // —2 a 4/1 
F(x) — £(0) — «<F’(0) — ro 0) a Tea Ee" ( 620). | 
Next, place 4” (6x) = F'”’ (0) + #, and then proceed as be- 


3 


fore, and so on, bearing in mind that the expressions a3 
eae together with their derivati to tl | » 
Ta34°°°) together wi r derivatives up to the (n—1) 


order inclusively, vanish for «=0; and we should have, for 


our final result, 


F(2)— F(0)—2F"(0)— Py F'O)—..) ge 

on Fay i gee) 
eT aly) ; 
whence 

2 
F(x) = F(0) + 2’ (0) + 5 F“(0) 
gril Nea a” ¥ 
si 1.2...(n—1) Yaga +333 nee (0a). 


And it appears that any real and continuous function F(x) of 
xis composed of the part 


F(0)+2F" (0) + 5 BAO oa 


geri 


+ 1.2.8...(n—1) 


(0); 


DIFFERENTIAL CO-EFFICIENTS. 79 


_ which is entire and rational in respect to a, and of the re- 


mainder 
gen 
Looe 
If the function is entire, and of the n” degree, its deriva- 
tive of the n” order will be a constant; that is, F"™ (6x) 


= F'" (0); in which case the development will terminate with 


FO (02 


the (n + 1)” term, and the remainder will be zero. For ex- 
ample, if #’(x)=(1-+ 2)", we oe have 


(1+ eP=14ne+n" ert... t+ nett a, 


61. The same principles Ne enable us to find the devel- 


opment of #(x-+h), arranged according to the seco C ee 
powers of either 2 or h.. For we have, Art. 56, 
F(e«+h)—F(x)=hF' («+ ah). 

Make hE’ («x +6h)=hF’(x) + R,, 
then 

Fi(a«+th)— F(x) —hF' (x) = | 
From this it is seen that /, 1s a function of both w and h, and » 
that it, and also its first derivative £”(a + h)— F’(x) with 
respect to hf, will vanish for h = 0: hence (Art. 56) 


F(a +h) — F(x) —AF' (x) = I” (a + 6,h) 


h? 
— 1.2 LECE) + J igh 
by making 
F" (2 + 0,h) = F" (x) + R;: 
whence 


oe 
Fi(a+h) — F(x) —hk’ (x2) — es Belo) ake 
and &,, together with its first and second derivatives with 
respect to h, will vanish for fa ei()ts Honan 


F(x+h)— F(x) — Ae 


FH (w+ Oh), 


80 DIFFERENTIAL CALCULUS. 


The manner of carrying on these operations is sufficiently 


obvious. We may write 


F(« +h)— F(x) — hE*(ar) ) 


h? 

ees h” n ° 

Dee en 1.2.3.6 

ATER ap ie ann 
whence 
F(a +h) = F(x) + AP (e) + 5 F(a) + 
h® 
+ (ORR as Fo i + a) 64) 


Tie in this last equation, we first make x — 0, and then, in the 


result, write x for h, we find 


F(a) = F(0) + wF"(0) + oe Fv(0)+... 


a” ; (2) 
T9355 


from which it appears that the formula of Art. 60 is but a 
particular case of that just established. But formula 1 may 
also be deduced from 2. For, in 2, change x into h, and make 
F(h) =f(« +h); then, taking the derivatives with respect to 
h, and in the results making h = 0, we have 

F'(h) =f'(@ +h), 

Fh) = fl (ae $ hy). 2. PO (6h) =e" ae 

£(0) =f (2), 

FM(0) =f" (2)... FO (0h) == fF @ (a aie 

But, when / takes the place of x, Eq. 2 becomes 


hn 

ee 
and in this, substituting the values of (kh), F(0),#’(0)... 
I" (6h), we have 


F(h) = F(0) +hF"(0) +... 


DIFFERENTIAL CO-EFFICIENTS. 8] 


f(e@+h) =F) WC MPa) +. 
ff (2 + Oh), 


which agrees with formula 1. 
62. When /(z) is such that the expression 


och 
1.2.3. 


for values of x between er limits continually decreases 


ete (0x) 


as 2 increases, then, by making n = o, ae formula, 


F(z) = F(0) + 2F'(0) +2 F(0) +... 


a” (n) 
7 ee ak ae ee) 


of Art. 61 will give rise toa converging series; and it may be 


written 
Oo) + 2 (0) 
Hee ie (0) 4 (1), 
which ia Maclaurin’s Formula. 


So also if #'(x) is such that the expression 


h” 
12.34. 2 Pu *) 


for values of x between assigned limits continually decreases 


as n increases, then, making n = o, the formula, 


Ea +h) = F(x) + hk" (x) + us Sh fed) 
paces 

iy RY 
of the preceding article may be written 
E(x +h) = F(x) + hl" (x) am 


‘ ‘1 


which is Taylor’s eee 
11 


+, EF (2 + Oh) 


82 DIFFERENTIAL CALCULUS. 


63. It may be shown that the quantities, 
a Dee 
122.3... oe Loe eee 


become zero when n is infinite. For take the expression 


m (nm +1) =m (a+ 1) —m? = am BEE m? 
CH) -Ci yeaa 
Pe ae a 

2 2 
which last form shows that the product m(m—m-+1) in- 


n+1 
2 


creases as m increases from 1 to ; that is, the product 


increases as the factors approach equality. This is also shown 


by the differential co-efficient 2 € ah — m) of the product 


taken with respect to m. Giving to m, in succession, the 
values 1, 2,3,..., the product will assume the successive 
values . . 

n, 2(n—1), 8(n—2)... (m— 2)8, (n —1)2; a, 
which increase from up to a certain limit, and then decrease 
by the same gradations down to n again. 

As nis the least value that this product assumes, the con- 
tinued product of these results, of which there are n, will be 


greater than n” ; that 1s, 
n. 2(1—1). 3(n— 2)... (n— 2)3. (n—1)2. n 
= (1.2.3. : (n—1)n) >n; 


whence 


Dene ear. fy < (Ga 


DIFFERENTIAL CO-EFFICIENTS. 83 


But, if x is finite, (<,) will be zero when vis infinite: hence 
n 


oe ay when nN —=0O; 
and the same is true of A” 
ait Denar 


Therefore it follows, that if F™ (0x), F™ («+ 6h), are finite, 
the products 


ac” | 
F (62), 
_ F (022), 


hn 
2d i SEA Ga) TAs 
posse. ape CE tle 


Meza A. 


will diminish without limit as m is made to increase without 
limit; and we can, in such cases, employ Maclaurin’s Formula 
for the development of /’(x), and that of Taylor for the develop- 
ment of /’(«-+-h), into series arranged according to the ascend- | 
ing powers of « for the first, and of either a or h for the sécond. 

64. Maclaurin’s Theorem, when applicable, may be stated 
as follows: The first term of the development of F(a) is 
- what the. function becomes when « = 0; the second term is x 
multiplied by what the first differential co-efficient of the 
function becomes when x = 0; the third term is the second 
power of « divided by 1 X 2, and this quotient multiplied by 
what the second differential co-efficient of the function becomes 
when x=0; and the (n+ 1)”, or general term, is the n” 
power of x divided by the product of the natural numbers 
from 1 to n inclusive, and this quotient multiplied by what 
the n” differential co-efficient of the function becomes when 
a 0. 

This theorem is of very general application for the expan- 
sion of functions of single variables, examples of which will be 
shortly given; but it is by no means universal: for 


1 
ee eae COL. 0, If — az , 


84 DIFFERENTIAL CALCULUS. 


are functions which become infinite when «2 =0; and hence 
the first term in Maclaurin’s Formula would be infinite, while 
the function for other values of x would be finite. There are 
other functions, such as y = ac? for which, though the func- 
tions themselves remain finite for « = 0, their first, or some 
of the following differential co-efficients, become infinite for 
this value of the variable; and, in such cases also, the for- 
‘mula would fail to give the development of the functions. 

65. Taylor's Theorem may be enunciated as follows: 
When a function (x +h) of the algebraic sum of two varia- 
bles can be developed into a series arranged according to the 
ascending powers of either taken as the leading variable, the 
first term is what the function becomes when this variable is 
made equal to zero; the second term is the first power of the 
leading variable multiplied, by the first differential co-efficient 
of the first term taken with respect to the other variable; 
the third term is the second power of the leading variable 
divided by 1 X 2, and this quotient multiplied by the second 
differential co-efficient of the first term; and the (n + 1)”, or 
general term, is the n” power of the leading variable divided 
by the product of the natural numbers from 1 to n inclusive, 
and this quotient multiplied by the n™” differential co-efficient 
of the first term. 

66. In Taylor’s Formula, the co-efficients of the different 
powers of the leading variable are functions of the other 
variable. When one or more of these functions are such, that, 
for a particular value of the second variable, they become in- 
finite, the formula fails to give the development of the origi- 
nal function for that value of the second variable; for then 


the function ceases to depend on the second variable, and is a 


DIFFERENTIAL CO-EFFICIENTS. 85 


function of the first variable alone, and will not necessarily be 
infinite for the assigned value of the second variable. 
For example, if we have 


P(x) =V/ x —a, 
then F(a thy=r/(«—ath). 
When «=a, F(x) =0, and the first and all the higher 
differential co-efficients of #'(x) become infinite for this partic- 


ular value of x; while, for this value, F(a +h) = Wh. 


It will be observed that there is a marked difference be- 


tween the failing cases for Maclaurin’s and Taylor’s Theorems. 
When Maclaurin’s fails for one value of the variable (x = 0), 
it fails for all; whereas Taylor’s may fail for one value of the 
second variable, but give the true development of the function 
for all other values. 

67. If a function becomes infinite for a finite value of the 
variable, its differential co-efficient will be infinite at the same 
time. In the case of an algebraic function, this follows from 
the fact that such function can become infinite for a finite 
value of the variable, only when it is in the form of a fraction 
whose denominator reduces to zero. But the denominator of a 
fraction never disappears in the process of differentiation : 
hence, if the function has a vanishing denominator, so will its 
differential co-efficient. In the case of transcendental func- 
tions, it is only by the examination of the different forms that 
the truth of this proposition can be established. Thus, in the 
logarithmic function y=Ix, y becomes infinite for «=—0; 


za _1 is also infinite for this value of x; and for the expo- 
ce. 2 

1 
nential function y = a”, which, if a>1, becomes infinite when 


86 - DIFFERENTIAL CALCULUS. 


BAA IhT) la 1 sores 
x —0,the differential co-efficient is mee which is 


infinite when « = 0. 

The circular functions tan. x, cot.x, sec. x, cosec. 2, which 
may become infinite for finite values of x, when expressed in 
terms of sin. 2, cos. x, are fractional forms to which the reason- 
ing in reference to algebraic functions applies. 

If a function becomes infinite for an infinite value of the 
variable, it does not follow that the differential co-efficient 
becomes infinite at the same time. 


Thus, in the example y = lx Whe : and y is infinite when 


’ dx 


Dre spite ce! = 0 for this value of a. 


dx 


68. It was remarked in Art. 62, that, unless #'(#) and 
F(x +h) are such that 


F(0) +2F" (0) +75 F"(0)+..-, 


; 52 
F(a) + LF! (a) +75 F” (x) + es 


give rise to converging series, the formulas of Maclaurin and 
Taylor will not serve for the expansion of these functions. 

A series in general is a succession of quantities any one of 
which is derived, according to a fixed law, from one or more 
of those which precede it. If wo, %1,&.,U3,...%,, are such 


quantities called the terms of the series, then we have 


Sn Uo tht Uy $ Ug +. s -Un 4 
for the sum of the first m terms. When this sum approaches 
indefinitely a finite limit S, as » continually increases, the 
series is said to be converging, and the limit in question is 


called the sum of the series; but, if the sum S, does not thus 


DIFFERENTIAL CO-EFFICIENTS. 87 


approach any fixed limit as n increases indefinitely, the series 
is said to be diverging, and has no sum. 
The geometrical series 


2 n 
a, ar, ar°,... ar, 


having ar” for its general term, has for its sum 
1—r” a ar” 
Pelee rr et. 7") = eh 
a Coren aa ) l—r 1—r 1-r 
It is evident that, as 2 increases, this sum converges towards 


a 


the fixed limit i if 7 is less than 1; and that, on the con- 


trary, as ” increases, the sum also increases indefinitely if r is 
greater than 1. 


We are assured of the convergence of the series 
Uo, Uy, Uy + © s Un, 
when, as 7 increases, the sum 


po ty 4 yp i yy 
converges to a fixed limit S, and when, at the same time, the 
differences 


Bee Dn — te; Data Bn = Ua t Uni aS T | 


vanish when 7 is made infinite. 

The limits assigned this work do not permit an investiga- 
tion of the rules by which, in many cases, the convergence or 
divergence of a series may be ascertained. 

69. Admitting that #(x) can be expanded into a series 
arranged according to the ascending integral powers of 2, 
Maclaurin’s Theorem may be demonstrated as follows : — 

Assume 

F(ez)=A,+4,07+ A,w?+...4+A,0? 


in which A,, 4,, A,..., do not contain x, and the exponents 


88 DIFFERENTIAL CALCULUS. 


a,b,c..., are written in the order of their magnitude, a being 
the least; then, by successive differentiation, we have 


E(x) = aA, x9 + bDAje""} -. .". 4 

EF" (x) =a(a—1) A;x** + 0(b—1) A,w? 7+. 
+p(p—1) 4,2? 

F'(x¢) = a(a —1)(a — 2) Ayx*— 14 3(b ee atest a 
+... 4+7(p—1)(p—2) AgmP=?. : 


The assumed and all the following equations, being true for 
all values of x, make x = 0; then, since £(0), #”(0), #”(0)..., 
would in general reduce neither to 0 nor too, we should have 

A,=£(0), a=1, 4,;=F"(0), b=2, 
EEO) B(0) 
Wig ee 123° oF 
F(x) =F (0) + 2F"(0) += 


bg 


which is identical with the formula of Art. 62. 

«0. Taylor’s Theorem also admits of the following simple 
demonstration when the function /'(#-+h) can be expanded 
into a series arranged according to the ascending integral 
powers of one of the variables with co-efficients which are | 
functions of the other variable only. 


Assume 


F(e+th)=f(x) +fi(~7) ht +f(xe)h?+...+f,(a) h?, 
and differentiate with respect to x, and also with respect to h; 
then 


GF@+M _ GO) Ki) ja 4 Kl) Val) py 
eM MMe dn" oa 
dF (a +h) 


To Vi (4) BO Hf (ae) RO ow es fn (@) AP. 


DIFFERENTIAL CO-EFFICIENTS. 89 


But /(x-+h) involves hf in precisely the same way that it 
it does x; and, if we place x +h = y, we have (Art. 42) 


dF(a+h)_aF(y) dy_ aFy), , 


dx Cyeye.  / Oy : 
dF(x+h) dF (y) dy dF (y) 
aoa = les 
dh dy dh dy 
dF(a+h) dF(a+h 
hence ee ) ae eee) 


that is, these differential co-efficients are equal for all values 
of # and h, which can only be the case when they are identi- 


cally the same. This requires that 


eee) spi - HO), 


e= 8, Alo)——, A) 


dis 
also, by making h=0 in the assumed development, we find 
f(@) =F (2); 
whence ipa 8) ae Lt! (ar), fy (8) os eae: 
therefore 
F(e+h) = F(a) +hF (0) +75 F(a) +. 


+75 a P” @). 


12 


SECTION VI. 
EXPANSION OF FUNCTIONS. 


V1. THE application of the formulas, demonstrated in the 
preceding section for the expansion of functions, gives rise 
to many important series, some of which we shall now deduce. 

1. If F(x): (14.2), then 

Fi(eysm(l- 2), 
F(a) =m(m—1) (1+ 2)", 
Fe-v (xz) =m(m—1)...(m—n+2)(1 + a)r—"tt 
FO (2) =m(m—1)...(m—n+1)(14+ 2)"; 
therefore (0) =1, £’(0) =m, F'”(0) =m(m—1)..., 
F°-) (0) =m (m—1)...(m—n+2); 
and hence, by Art. 60, 


: (1-+ay=1+me+ mary 


(m—1)...(m—n+2) 
.(n—1) : 


—n+1)a” bee, 
1) 058 goes (Lop eee 


When ~ is less than unity, the last term in this development 


will diminish as m increases; and, by making 1 sufficiently 
great, the series 


1+ mem ot 4m UE “ty 


90 


EXPANSION OF FUNCTIONS. 9] 


will approximate more and more nearly the true value of 
(1+ a)” the greater the number of terms taken. 


2. Let (ge) ——.e7. Then 
Bee) = eee emer ee eg i) (ot) Ae ee 
eee 0 (0) = OY (0); 
F (6a) =e 0%: 
Pct eres c* 
therefore é Sway gt oak 
ihe xe” vx 
Bitiao, Seer el) To a ee 


Making in this « = 1, we have 
= gb ge a ee sala Sieg 
on Low pages 


a series that may be used for finding the approximate value 
of e. 


38. Let F(x) =sin.2; then 
a y—= COS. @ —- STi, (2 + 5 


d sin. (2 oe 2) 
F" (x) = See" — cos. (2 a 5 sim (2 ote i) 


Qn 
d sin. (2 = a 
- y} oe Qn ae 37 
FE (x) Pa a pe —= COS. (2 + e) sano USO (2 + a) 


| ey — Sin; (2 ao 7) 


Bete On (Oyo 1 FY (0) —- 0, Fh" (0) = — Te 


Therefore 


F-) (0) = sin. dese as 
and we have 


92 DIFFERENTIAL CALCULUS. 


; oe ac® 
sin. & = ®— 754 +79 Re ea 


erg Mego of - n—d 
TL28.c(n = 1) 


we” ; nn 
Ta 28 es (00-475 
4, Let F(a) = cos. « ; then ' 
I" (x) = — sin. & = cos 2 -E 5) FY" (a) = cos a ap 


zat 


fu (x) prem 16). (2+3) ae (x) — cos. (2+ 3) 
LEO) = eae) = 07h 0) — 1, F (0) 0% ih 


pa” rain ) c= COR. s mn: 
ac? act 
h Pia | ee ae 
Seem oy ake van oe 
ont n—1 


vf 


AM 


at nn 
T4508 eae G ay =) 
By Art. 63, it will be observed that the last terms in Exs. 


2,8, and 4, diminish as 7 is increased, and finally vanish when 


nm becomes infinite. 


5. Let #(x)=/1(1+ 2); then 


/ aes 1 /1 — 1 V1 oe 1.2 eke 
ea i (2) — eee (7) (aaa f 
Bae) (ar) = es rt Oe Gaia): hence 


E(0) =.0, 2! (a) = 1, 2 (00) I ee 
Bo (0) = ( — 1)*- 11.2.8... (n— 1); and thererane 


2 3 4 es a 
(—)* be 


a nm ° (1+ 62)" 


EXPANSION OF FUNCTIONS. 93 


An examination of the last term of this expansion shows, 
that, when x does not exceed unity, this term necessarily de- 


creases as ” Increases, and vanishes when n becomes infinite. 


And, since the factor aoe ; under this hypothesis cannot 


exceed unity, the sum of the series, up to the n™ term in- 


. . 1 
elusive, cannot differ from the true sum by more than—; and . 
n 


hence, by increasing 7 sufficiently, this difference can be made 
as small as we please. 
Changing the sign of x, we have 


a 2 bistemte.. ee a ganl 
(= 1 yes a” 
1) (1— Ga)" 
6. Let | Pe) —— tar, ee then 
Hee ey = E 5=5( cee 
Ite 2\l—aV/—-1 1ltae/—l 


1 ee gore 
=3(a-ev=1)> 4c +av=1y") 
P(n) =5 x ee sere AY AT 8 aes ies 
1 avi Soatt 
a I TE I a a ie 
tes 1 gees! ee 
=V=1x35(a—ev=1)*—(- 1a ev=1)") 
L(g) 
Be UN 


ult 2 shal iis 
1) ee yi) 1 (L+2V—1) 
Ae SP itor ite: ee 
ea) oA) ea) (eV) (— yee 2)" 


94 DIFFERENTIAL CALCULUS. 


therefore 
ah 12.3.6: ( be 
PO =v) 
Whence it follows, that, if n is an even number, #"™ (0) = 0; 


but, if m is uneven, then 
n—1 1.2.3. .(m —1) . 


FM (0) = (= 1) 5 2 
EY 103) (na + 
/—1 /—1 
Hence we have 
3 5 n—1 
gel te a ee po 
taney aso Acres teas med 
eee (1—oar/—1) "(1+ 1)". 
n Qr/—1 


The final term in this development is not in a convenient 
form, as it stands, to decide whether the series is converging 
or diverging; but by referring to Ex.-18, p. 67, making 


a = 1, and observing that there 6 = 5 — tan.'x, we have 


2 —— 
Fo (a) Oe (= 1)*71 19. Chena (n ’ 1) sin. er _- ntanmte \: 

(1 + a”) y 
therefore 


3 5 
tan“ e =a — = +5 —& 4+... 


(— eee ie fe sin. (s — ntan.! ® 
(12%)! ae 

This form of the final term shows, that, if 2 is less than 
unity, the numerical value of the term may be made as small 
as we please by giving to 7 a value sufficiently great. 

The above form for /"” (x) might have been used for find- 
ing all the differential co-efficients of tan. a as readily as that 
specially deduced for that purpose. 


EXPANSION OF FUNCTIONS. 95 


The following is a more simple process for getting the 
expansion of tan.~! # : — 
Assume 
tana =— A+ Be+ Cx? + Dx?+ ke. (1), 


and differentiate both members with respect to x; then 


1 2 
eee wet Ae (2); 
but by division, or by the Binomial Theorem, 
aes —xvtoat—a>te*§—&+ ke. (3). 


The second members of (2) and (3) must be identical: hence, 


equating the co-efficients of like powers of x, we have 


1 
Je femty ie Gia. 0; eat I ies ATA ee 
and, since the assumed development must be true for all values 
of x, make x = 0 in (1), and we find 4=—0: therefore 


7 
— Wo ee 
7 


tan.-!'2%—a—- 
3 et 


feat —— sin, ' 27, assume 
sin. ‘w= 4+ Bet Cx? + Dx? +... (1), 


and differentiate both members; then 


1 ; 
OO) a Si ee te ae la ace). 
but, by the Binomial Theorem, we find 
1 1 1.3 letra 


Trial tar tag t+agaee t+ (3). 


The co-efficients of the like powers of w in the second 


members of (2) and (3) must be equal: hence 
i Loa) 
B-1,0=0,D= 55, #= eae boa | 


and, by making x = 0 in @) we get A = 0: therefore 
Tends) Val oso ce 
a es 


ney | x 
a raat 545 12467 


96 DIFFERENTIAL CALCULUS. 


8. Let y=e* "= and assume 
y=A,+4,24+4,07?+...+4,0"+... (1). 


Differentiate twice; then 


oy 2A4,12.34,¢-+. + (n—1)n4, 0" 7 (3). 
But 

dy asin T zh et ee 

daz Ji — 2? 

ary — prsin te a? | See 

da? Le (1 — a)? 
and hence 


(1—a?)"%— a7 Faary... (4). 
8b 


dy d*y 
Substitute in (4) the values of —*, —“, taken from (2) and 
hae Gaee 


(3), and we have 
24,+ 2.34,0 +3.44,07+--+1(n—1)n4,0"74+.. 
— (24,0”°+2.34,2°+3.44,2t+--+(n—1)nA,x"+..) 
—( 4,*%4+24,0°+34,0'+4A,v'+. +. i 
a? A,+a’?d,x-+a?A,x? Be a 


tet dye" 


Equating the co-efficients of the same powers of x in the 


_ two members of this equation, we find 


eas 
and generally 


2 2 
Bows. 4 2 Uh on 


Yael ind Areal SIN 
(n—1)n 


EXPANSION. OF FUNCTIONS. 97 
If then A, and 4, be found, formula 5 will give all the 


following co-efficients in terms of these two. 


. - —l . 
A, is what e**" * becomes when «= 0: hence 4, = 1. 


~ becomes whena = 0: 


Jt 


And A, is Snakes — e@ sin) —___ 
dx 
hence 4,= <a. 
In formula 5, making n equal to 2, 3,4, &c., successively, 
we get 


ia 
A= Ty 4 Lis 


ae Eadealkeg Vpok + (@? 4-27). a? 
F291 2.8.4 


Substituting these values of dy, 4,,A,...in the assumed 
development, it becomes 


SW ara, al(ar--l1 : 
eee ae 


a(a? +1) (a?+ 87) . 
2 nana aaa a 


By Ex. 2 of this article, we also have 


hoes re 9 odd 
ea sin. falfasinét e+, (sinet2)?+5 5} (sin 'a)* +. 


Equating the co-efficient of the first power of a in this 


series with that of the same power of a in the preceding 


series, we have 


Lz? | ek ero ca sl BRS ies ers od 
;peenae | 
ee epee AION AG OT ath 


as in Ex. 7. By equating the co-efficients of a?, we should 
also find 


92 ari cals ee vO" 
Eee ae r tI : : 
eee ed” 1515.67 3.456.738" 


8 


x 
peep L(.1 -~ ery, ee iy tag 931 5 51 ee 


13 


98 DIFFERENTIAL CALCULUS. 


10. "y =I oe fie \ig et et Se 
10. y=l1—a+e?) y= —aot 5 + 5 may oe 
.2 3 
ll. y=1(1+sin. 2), y=u— +a... 
ey 0? 3 ; Ta4 
AD: Oe eee s Yl 2+ > 


NS Be 10\ 1 
13. ty=(“ wae z) yr show for what values of « 


Taylor’s Theorem fails to give the development. 
It fails for 7 —c; 1st term is then infinite. 
It fails for «=a; 2d differential co-efficient is then infinite. 


SECTION VIL. 


APPLICATION OF SOME OF THE PRECEDING SERIES TO TRIGO- 
NOMETRICAL AND LOGARITHMIC EXPRESSIONS. 


72. Leva and b represent any two real quantities what- 
ever; thena+b*/— 1 will be the most general symbol for 
quantity, since, by giving to aand b suitable values, it may be 
made to embrace every conceivable quantity, real or imaginary. 

The two expressions, a + bW — 1,a—bW — 1, which dif. 
fer only in the signs of their second terms, are said to be conju- 
gate; and their product, (a + bW — 1)(a —bW —1)=a?+2?, 
is always real and positive. The numerical value of the square 
root of a+ b? is:the modulus of either of the conjugate ex- 
pressions. Denote this modulus by rv; then it may be shown 
that the expression a + b*/ — 1 can be put under the form 

| 7 (cos. 6 + / — 1sin. G). 


Horie, @=7 cos. 0, 6 —rsin. 6: 
b ; 2 
Ha tan. 0 =~, r? (cos.? 6 + sin? 6) =r? =a’?+ 6’, 
r=Va? + b?, 
Now, if we suppose the arc of a circle to start from—Z, and 
to increase by continuous degrees to + 5 passing through 


zero, the tangent will at the same time increase by continuous 
degrees, and pass through all possible values between — a 


and +-2». Among these values of the tangent, there must.be 


one that will satisfy the equation tan. 6 = ie and the arc an- 


99 


.100 DIFFERENTIAL CALCULUS. 


swering to this tangent will be that whose sine and cosine will 

satisfy the equations a —=~rcos.0, b=rsin.0, and therefore 

render 7 (cos. 0 + / — 1 sin. 0) the equivalent of a+ b7/— 1. 
73. Let us resume the series (Art. 71, Exs. 2, 3, 4). 


2 3 
e=ltoetoat+ypogt 


F a8 a? 
Bl, B= 2 — Tog to 34 5 
a? oot 
Ey seers Gee EN ak MMS ek ee bes I 
soa y 1277234 Gh: 


and in (1) write «WV — 1, —a~/— 1, for a successively; then 


x? eirn/ —] a" 


eVE1 Fay eecenee 

SOP lbey: i737 ae 
ei a/—] mae 
heap gag ch = C08: acta ae 


as is seen by comparing this result with the second members 


of (2) and (3). 


Also e-*¥-1 =1—avW—1— 


1.2 L255 1.2.3.4 
PV Aas, | poll. 
ie che = 00s. &— 7 — 1 sin. @: 
therefore cos.a+%—Isin.2=e7%—-!.-- (4) 
cos. ¢ — 4/— 1 sin.@ = e7*~~1s =. (5), 
also cos. ¥ + 4/— 1 sin, y =e") ae (6); 


multiplying (4) by (6) 
(cos. a -E A/T sin, x) (cos. ¥ + /— 1 sin. Tie er+yy=1 
= cos. (a + y) + Y— Lsin. (a + y). 
Effecting the multiplication in the first member, and then 


equating the real part in one member with the real part in the 


TRIGONOMETRICAL EXPRESSIONS. LOE? 


other, and the imaginary part in the one with the imaginary 
part in the other, we find 
cos. (x + y) = COS. & COs. y — SIN. XSiN. ¥ 
sin. (w + y) = sin. «cos. y + sin. ¥ Cos. x, 
Again: 
(cos.x-+4/ — 1 sin.) (cos.y + — Isin.y) (cos.24- — Isin.z) 
= ettyts...)Y-1— 98. (at ytet-.)+v —Isin.(etytet-.), 
from which, by making «x =y—=z=.---, we have 
(cos. @ + 4/ —1 sin. a)” = cos. max + / — 1 sin. mx, 
and generally 
(cos. g + 4/— 1sin. a) ™— cos. ma + / — sin. mz, 
which is known as De Moivre’s Formula. 

Hence the multiplication of expressions of the form of 
cos. 2 + 4/ —1sin.ax, and therefore of all imaginary expres- 
sions, is thus reduced to an addition, and the raising to 
powers to a multiplication. 

V4, Dividing formula (4) of the preceding article by (5) 
of the same, member by member, we have 


a cos.a+7—Lsin.2 sol-b Man, 


ee —1 cos. ee a I sin. x 1 — MrT, 1 tan. nee 


whence, by taking the Napierian logarithms of both members, 
9a /—1=!1 (1 + /— 1 tan. a) af (1 —/— I tan. 2). 


Expanding the terms in the second member by Ex. 5, Art. 71, 


Sow tate tanta 
9n/—1=r*/—1tan.2+ Day —V-1- aes “ a 
SE EA ey Rag ee 
pass 
eae x 
—(- S/o 


tan.! x Ss tare? x 
a rea ore te ot : ») 


‘102 DIFFERENTIAL CALCULUS. 


Equating the imaginary parts in the two. members of this 
equation, and then dividing through by 2/ — 1, we have 


tan’a , tan®a . tan.te 
ome Bat. their 


a series that may be used for the calculation of z, and which 


2— tan. x — 


agrees with the formula in Ex. 6, Art. 71. 
4". To find the expansion of cos.” in terms of the cosines 


of multiples of a. 


Make ee" Fl ay? then er, ) ey ee 


eame/—1 — iby 
Yt 
From formulas 4, 5, Art. 73, we find 7 
= = 1 
2.cos. # = e*~—! + e7#v-1 — y + 
oe : 1 
24 —1sin.2@ = et! —. (Te 
also, from De Moivre’s Theorem, we deduce 


1 press 1 
2cos.ma = y™ + yn 9/—1 sin. ma = y™ — re 


Because eowe=y +i, 2" conn (y +=) 
a 


but 


Va. 


yr it. Ae 


1 ‘ n n—2 
(+5) BY CUS Solan ee 
De tate 1 1: 
Ringer Fi 3 pat pat oe 


ay pat uy n—2 1 
=a" boat (vee 


= +, 


iar lee 
tal: 


by combining terms at equal distances from the extremes: 


hence 


TRIGONOMETRICAL: EXPRESSIONS. 103 


n — i | 
CORN T= | COB: Mee +n cos. (n — 2)x 


n— 


cos, (n—4)xe4+.-. ) (b). 


Since there are n + 1 terms in series (a), when 7 is even, the 
number of terms is odd, and the middle term, that is, 


n(n —1)(n—2)... (942) +1) 


? 
pete fe 
2 9 


will be independent of y, and consequently of #; but, when n 


+n 


is odd, n + 1 is even, and there is no middle term in series (a), 
and therefore no term independent of x. In the first case, 
there will be within the ( ) in formula (0), besides the term 


n sy» 
that does not depend on a, 5 terms, containing as factors the 


first cos. nx, the second cos.(m —2)x; and so on to the last, 
which will have cos. 2 for a factor. In the second case, that 
is, when 7 is odd, there is no term within the ( ) in formula 


(b) that does not involve x; but the iz x : terms will then have 


for factors, severally, cos. nx, cos. (m — 2) xa. .., cos. 3a, cos. x. 
eae eas 26 = i (cos. 4a + 4 cos. 2% + 8) , 
1 
Ex. 2. cos. et — 3 (cos 5a + 5 cos. 8a + 10 cos. *) : 


76. To find the expansion of sin.”x in terms of the sines 
of multiples of a. 
By formulas 4 and 5 of Art. 73, we have, employing the 


notation of the last article, 
2/—Isin. 2 = et¥—! ey aie ere) deny 


a 


104 DIFFERENTIAL CALCULUS. 


aes Rea Nes eae se 
therefore 2” (Vf — L)* sin. = 2" (= 1)7 ee (y be. -) 


n—1 


att ee n—2 ms 
= ny +n 9 


y™*— &+- ae 


p —1 1 Fs 1 al 
RGM Mp G eer 


ge 
1 1 
i n ee if n na n—-2 _ poe | n> 1S ey 
=(y een) a) n(y pec: mi) 


n(n — 1) n—4 n—2 1 
is 130. (y ie 7) a he (a). 


An examination of tlus series shows, that, when 7 is even, 


the second terms within the ( ) are all plus; and, when v is 
odd, they are all minus. In the first case, the expansion of 
sin." will involve only the cosines of multiples of #; and, in 
the second case, it will involve only the sines of these multi- 


ples. 


n 


The factor (—1)? in the first member will be positive and 
real when 7 is any one of the alternate even numbers:begin- 
ning with 0; that is, when m is 0 or 4 or 8 or 12, &.; and 


negative and real when v7 is one of the alternate even num- 


bers beginning with 2. In like manner, (—1)? will be imagi- 
nary and positive when m is any one of the alternate odd 
numbers beginning with 1; and it will be imaginary and nega- 
tive when 7 is any other odd number. 

Let & represent any positive whole number, zero included; 
then the different series of values above indicated for n will be 
embraced in the four forms, 4k, 4h + 2; 44 +1, 44 +3. 

It would be of no advantage to make formula (a) conform 
to each of these cases by special notation, as it can be easily 
applied, as it now stands, to the examples falling under it. 


TRIGONOMETRICAL EXPRESSIONS. 105 


Ex. 1. Expand sin.’a in terms of the sines of the mul- 
tiples of x. 
22 (a/ 21) sina (y' — 3 — 3 (y - 
y" y 
= 2/— 1 sin. 3x — 6 —1 sin. 2: 
1 
Aap sin32a2 = — rr (sin. 3a — 3 sin. @). 
Ex. 2. Expand sin.‘z in terms of the cosines of the mul- 
tiples of x. 
7 oe 1 1 
24(4/—1)‘sin.tc = (y'+ 7) —A4 (v"+ =) +12 
= 2 cos. 4% — 8 cos. 2a + 12: 


hse 52 (cos 4x — 4 cos. 2x + D) 


Ex. 3. Expand sin.x in terms of the sines of the mul- 


tiples of x. 
2? (4/—1) sinc — (y— = — 5 Cis 7) + 10 (y _ ) 
= 2/—I1 sin. 52 —10/ — 1 sin. 3u + 20 /— 1 sin. a: 
; Ie : 
sin.“ = ry (sin, 5x2 — 5d sin. 3x2 +10 sin. =) 


Ex. 4. Expand sin.*a in terms of the cosines of the mul- 


tiples of a. | 
—\_ . 1 1 1 
Day — 1)*sin* a — (+ _ — 6 (y+ a) +15 (y+ — 20 
y° Y y° 
= 2. cos. 6% — 12 cos. 4x + 30 cos. 2x — 20: 
ee — sal 08 6x2 — 6 cos.4%-+ 15 cos. 2% — 10), 
77. To find the different n™ roots of unity. 


Let x represent the general value of the n™ root of unity; 


then, by the definition of the root of a number, «”=1, or 
14 


106 DIFFERENTIAL CALCULUS. 


2” —1= 0; and the object of the investigation is to find all of 
the values of x that will satisfy the equation x” —1=0. 
By De Moivre’s Theorem, Art. 73, we have 
(cos. y + /—1 sin. y)”= cos. my + /—1 sin. my; 
an equation which holds, whether m is entire or fractional, 


positive or negative. Now, if k be any whole number, 2ka — 


will be an exact number of circumferences to the radius unity, 
and 
cos. (y + 2kx) = cos.y, sin. (y+ 2k) = sin. y: 
therefore (cos. y + /—1sin.y)” 
= (cos. (y + 2kx) + /—1 sin. (y + 2kn)) 


1 ee 
Make m = 73 then (cos. y + “/—1 sin. y)” 


a (cos. (y + ka) + of —1 sin. (y + 2k) 


2h pesae 2k 
Yee re emrs i 
n 


== COS. 


In this last equation, make y = 0: whence, as cos. 2ka= 1, 
and sin. 2kz = 0, we have 
2kn 


1 2k ue 
(1)" = cos. —~ + /—1 sin. 
n n 


1 
But, from the equation «”*—1=—0, we get x=(1): hence 
we conclude that the different values of x, or the roots of the 


equation x” —1= 0, are the values that may be assumed by 
2h att 2h 
cos, -— + /—1 sin. — 
n n 


by assigning different values to k. Since k may be any 


whole number, take for it successively 0, 1, 2, &c.; then, 
1 
when ‘amet US 1" = cos. 02 '4/ 1 sins Oe 


1 2 ee 2 
when Derek 1. 08: — + 4/—1 sin. ~ 


ee 


TRIGONOMETRICAL EXPRESSIONS. 107 


L 4 basa 4 
when ae 1" = cos, = + /— I sin. 


‘3 
and so on, continuing the substitutions for & until the 


arc reaches such a value as to cause the expression 


2hr — . kx 
cos. —— +k 4/—1 sin. ae to reproduce the roots it has already 


given. When n is an even number, this will be the case 


for k = 5; for, . 
n z Tiny, at eT eo 
if Bohs iB Reet tafaley at 4/—1 gin. nu; 
n .: er, 
if Liaary 1? cos. 2 AW 1 gin hs ST 
n u n+ 2 eee ie 
if a rl, 1 COs: gt + 4/ —1 gin. nt; 
— 2 pee 2 
but Cos. “ git / —] sin. — nm 
v 
2 een’, 2 
_) SaeaY te iat aN a mt: 


11) 


therefore the two roots corresponding to k= 5 + 1 are the 


same as those corresponding to k = 5 —1. So, also, those ob- 


tained by substituting 5 + 2 for k are equal to those obtained 
n 
2 


tions for k after the value 5 would merely reproduce the roots 


by substituting 2 for k, and so on: whence all substitu- 


already found. 


Again: when 7 is an odd number, the substitutions for & 


must be continued until k = al ; for, 


108 DIFFERENTIAL CALCULUS. 


1 
n 


RT ie eon od Coeds ma ra 


if Lan eet Pemee ep Paine 


my 


but cos. one 4A OT eine tee Ve = és a 
n n n 


=e 4/ — Lsin. 


te 


n+] 
n 
hence the substitutions of ~ aa na 2 ait : for k give the 


same roots. So, also, it may be shown Re the substitutions of 


Lees Pande 


2 


for k would give the same roots. Therefore 


we should merely reproduce the roots already found, if we 


substituted values for k greater than k= m—1. 


2 


When n is even, k =0andk =5 give, the first the root 


+1, and the second the root —1; and the intermediate 
values of k give each two roots. When n is odd, k=0 
—1 
2 


inclusively, give each two roots. In either case, the expres- 


gives the root 1; and all the other values of k, up to i 


Qhrn Qn 
sion cos. —— ++ 4/— | sin. —— can assume vn different values, 
| n 


and no more. Hence it follows that the equation x” — 1= 0 
has n different roots, and can have no more. 

By the aid of the foregoing principles, the roots of the 
equation «” — 1 = 0 may be expressed under the form of ex- 
ponentials. 

Since, by Eqs. 4,5, Art. 73, we have 


Soe JES 
cos.2Ea/—1sin. e— ert” ’ 


TRIGONOMETRICAL EXPRESSIONS. 109 


the successive values taken by the expression 


hn a. : Qlen 
cos. — + 7 — | sin. — 
n n 


may be represented in order by 


eet AL arial 
€ ,e ” a ts #5. € ; 
when ” is even, and by 
Pye ot 8 yo, tS el 
€ ay rT taal stacotge Curti® ) 


when is odd; the first term in each series of roots being 


unity, but the last term in the first series is minus 1, since it 


is equal to cos.7 + 4% — lsin.w—=—1. Both series of roots 
are the terms of a geometrical progression, the first term 
cl sei 


+0/—1 : . ° . 
= 1, and of which the ratio is e 


of which is e~ 
Ex. 1. What are the three cube-roots of unity? 
They are the roots of the equation «*— 1=0. 


Here n = 38, and the proper values for & in the expres- 


sion Cos. ass aA/ — 1’sin, a are (0 and 1: hence the first 
gives 
qa)" — cos.0 + /— 1 sin.0 et 1 =1. 
* The second gives 
5 gas 


1 


ae Q7 Bags. LET, 
(1) = Bosaeeet A/S sin. =e 
Ex. 2. Find the roots of 7° —1=0. 
Here n = 6, and the proper values for & are 0, 1, 2, 3. 
(1)8 —cos.0 + V — 1 sin. 0 = coger heey 
, It c2e ee mu ae 
(1)8 = cos. g + VY — 1 Sling Sac a 
bee eae ae) re ga 
(1) = cos. Sf /—1 sin. Baga 


+rr/—1 


(lye = cos. 4/1 sin. x = e+ Key. 


110 DIFFERENTIAL CALCULUS. 


For each root of the equation «” — 1 = 0, there is a binomi- 
al factor of the first degree with respect to # in the first 
member of the equation. Since k=0 gives but one root, 
unity, there will be but one corresponding factor z—1: k=1 


~ gives two roots, and the corresponding factors are 
| Zee 2 ne! 
2 — (cos. Zid isin.) © — (cos. St — 7 Tain, 2) 
n n n n 
which by multiplication will produce the quadratic factor 
x” — 2x cos. a +1. 


In like manner, each pair of simple factors may be reduced to 
a quadratic factor. If is even, the last factor is « + 1, which 
may be combined with the first factor « — 1, producing the 


quadratic factor «7 —1. Hence, when 7 is even, we have 


xn — I =(2" = 1) (2 — 28008, + 1) (2 — 2x roma 1) 
n n.* 


att) 
and, when n is odd, 
a —1=(e—1(2 — 2 eos. = +1) (22 — 22008, +1) 


‘chanel 1} 
Tex, he (CL) = 1)(2* — 2x cos. + 1). 

Ex. 2. («#*—1) 

= ts — 1) (2 — 2x cos. a 7 (2° — 2x cos. ae 1) 


78. The solution of the equation «” + 1 = 0, and the reso- 


lution of its first member into factors. 


Nn 
: (2° — 2 Gos. 


Nn 
; (2° — 2a cos: 


Resume the equation 


] 
(008. y Tain, y)* = 000. LO 5 5/ 


TRIGONOMETRICAL EXPRESSIONS. 111 


of Art. 77, and make y=; then, since cos.2 = W— 1, and 
sin. z = 0, this eS becomes 
1 


(==) 1)\=='co tee I 1 sin, Bea, 


But, from «” + 1=—0, we have x = (— 1)*; hence the roots of 


the equation 2” + 1=0 are the values of which the expres. 


sion Cos. ae + / —1sin. a ce Ya will admit for ad- 


missible values of k. But k may fe any whole number in- 
cluding zero. Therefore take for k successively the values 
(leer then, 


1 
fork=0, (— 1)"=cos. § +7 — 1 sin. ; 
n n 


1 
fone ty (—— L)* = cos. 2 4 I sin. 


a min aie ena 


for k=5— in ie = cos. 

When 1 is even, substitutions for k greater than 5 —1 will 
only reproduce preceding values for (— 1); for, if k= 5 
then (— 1)" a cos. (x i) aay Tid (x + 7 


a=s7 COS, (« — ‘\ev=a sin. («— 
n n 


which is the same pair of roots as that given by the substitu- 


tion of os 1 fork. In like manner, it may be shown, that, if 
n ; 
*S5 +1, the pair of roots would be the same as that for 


b= 52; and so on, 


14194 DIFFERENTIAL CALCULUS. 


‘When n is odd, the substitutions for k must be continued 


stil eae 

if k=" Z 3, (— 1) = 008. "ae =I sin.” fe 
: n—t1 r ‘ 

Vey = ,(—1)"=co. 2+ 1sin.z= — 1, 


Now, for the next value of k, that is, 4 = are Ee) | jee 


1 
(—1)*=cos. 1 * i ea 1 sin. Tee 


2 


n—2 ere 
== 008: te / — 7 sind ee 
n 


and therefore this substitution for k gives the same pair of 
Stee 
roots as is given fork = am °, and the higher values of k 


merely cause preceding pairs of roots to recur. | Hence, 


whether 2 be even or odd, there will be n, and only n, differ- 
‘ : 
ent values for (— 1)”; and the equation x” + 1 =0 has n, and 


only n, different roots. These roots can be put under the form 
of exponentials, as in the case of the roots of #” —1=0. 
Ex. 1. What are the roots of «'+1=0? 


Here n = 4; and the formula 


iL a ad 
(—1)*= cos, Tht VI i ir 
n 


e mt So i eee 
gives, for k =0, (— 1)» = cos, 7 + VW — 1 sin. 73 


for ieeoek, fect Na EE om = Isin, ve 


TRIGONOMETRICAL EXPRESSIONS. 113 


Ex. 2. What are the roots of «° +1—0? 
Here n = 5; and the formula gives, 


1 
fork—=0, (—1)" = cos. 7 + Vv —Tsin. 5 


om , 


ay 
Mites, (—— 1) cos. 2 + 4/—1 sin. -? 


1 
for k = 2, (eee Conte A 20 sin. wie) 1. 


For each root of the equation x” + 1 =0, there is a corre- 
sponding binomial factor of the first degree with respect to x 
in the first member of the equation. 

When v is even, all the roots enter the equation by conjugate 
pairs, and the factors of the first member, answering to the 
simple roots of each pair, may be compounded into a rational 


quadratic factor, and we should have 


o +1—(a4— 20 cos.= +1) (2 — 22 008.5% 4-1). - 


(2 = 22008." Sa +1), 


tv 


When 7 is odd, there will be rational quadratic factors for 


IO 


eww |. ..,up to k= inclusively; but, for k= 


n —I1 
2 
case, we should have 


ot + 1=(2*—22008,% +1) (2-22 cos, 1): “se 


, there is only the simple factor x +1; so that, in this 


n—2 
(22 — 2x 008, x+1)(e +1), 
The solution of the equations x” —a—0, «"+a= 0, and: 


the resolution of their first members into simple and quadratic 
15 


114 DIFFERENTIAL CALCULUS. 


factors, may be at once effected by the formulas in this and the 
preceding articles: for these equations give respectively 
Le o de 
2—a"(1)", «—(a)"(—1)*, 
in both of which na is the numerical value of the oe root of a; 


and this, multiplied by the different values of (1), will give 
the roots of «”—a=0; and, multiplied by the values of 


(— 1), will give the roots of «”+a=0. 

79. The determination of a general expression for the log- 
arithm of a number positive or negative. 

In any system of logarithms, the logarithm of 1 is 0, and the 
logarithm of 0 is — if the base is greater than unity, and 
+ o if the base is less than unity; while the logarithm of oo 
is + 0c or— o, according as the base is greater or less than 
unity. It thus appears, that, whatever be the system, all pos- 
sible positive numbers between 0 and will embrace for their 
logarithms all possible numbers between — oand +. The 
logarithms of negative numbers, if they admit of expres-— 
sion, must therefore fall in the class of imaginary quantities. 


In the equation 
cos.% + 4/—1sin.a=e*~—! (Hg. 4, Art. 73), 
write « + 2kz for x, k being any whole number; shen 
cos. (a + Qk) + 4/—1 sin. (a + Qhkw) = e@ +27) Y=1, 
Por 2.0, this pives.-1= ¢#*"—- 
for «= 7; this gives —1—=e@t+07~—-1, 


Taking the Napierian logarithms of both members of these 
equations, we have 


L(1):== Qh A/T, (1) = (2k + 1) Sn 


LOGARITHMIC EXPRESSIONS. Ls 


These are the general expressions for the Napierian logarithms 
of 1 and —1: and, since k may be any whole number, it fol- 
lows that both +1 and —1 have an infinite number of log- 
arithms; but all of them, except that of +1, corresponding to 
k = 0, will be imaginary. 

From this it may be shown, that any positive or negative 
number, in whatever system, has an indefinite number of 
logarithms. | ; 

For, first, suppose y to be any positive number, and « its 
arithmetical logarithm taken in the Napierian system ; then 

Tg em*—et*x1=e*~x euavy—i net ertuny—1 . 
ae y= = x + 2ha Mel 
which is the general Rote ees of y, and will ad- 
mit of an unlimited number of values. Denoting the arith- 
metical logarithm by /(y), we have 
ip Uy) 2k —1.. > (Mm). 

Again: let Ly denote the general logarithm of y, taken in 
the system of which a is the base, L(y) denoting the arith- 
metical logarithm; then, since we pass from Napierian to any 


other logarithms by multiplying the former by the modulus of 


the system to which we pass, multiply Eq. m by _ the 
a 


modulus of the system characterized by LZ, which gives 


1 1 Apa oa 
ly X yee L(y) X ia + ig 2h —13 
or, 
2ht 4/1 
eat a 5 (2). 
a 
From Eqs. m, n, we conclude that the arithmetical logarithm 

of a positive number taken in any system is the value of the 


general logarithm corresponding to k = 0. 


116 DIFFERENTIAL CALCULUS. 


Now, suppose y to be negative ; then —y = — 1 X y, and 


xX YO ore bb, paw Leen e@k+lav—1 a ettQk+l)rV—1 . 


l(— y) =v@+ (2+ 1)aV/—1... (p), 
_&-+(2kh+ lav—1 (q). 


la 


also L(—y) 


Eqs. p, q, are the general expressions of the logarithms of a 
negative number, and show that such a number has an unlim- 
ited number of logarithms, all of which are imaginary. 

From the equation 1(/—1) = (2k+1)aW —1, we get 

pio el gel 
(2k+1)/ —1 


This and the preceding remarkable results developed in this 


section must be interpreted with reference to the symbols 
and the character of the quantities with which we are dealing. 
It must be remembered that e and w are the representatives 
of arithmetical series, and that the formulas have meaning, 
and can be regarded as expressing true relations, only when 
the rules for combining imaginary quantities with each other 
and with real quantities are strictly observed. 


SECTION VUI. 


DIFFERENTIATION OF EXPLICIT FUNCTIONS OF TWO OR MORE IN- 
DEPENDENT VARIABLES, OF FUNCTIONS OF FUNCTIONS, AND OF 
IMPLICIT FUNCTIONS OF SEVERAL VARIABLES. 


80. WHEN several variables are involyed in an equation, 
any one of them may be selected as the function or dependent 
variable; the others being regarded as independent. If the 
value of the function is directly expressed in terms of the va- 
riables, we have an explicit function of several independent 
variables; but, when the function and the variables are in- 
volved in an unresolved equation, we have an implicit function. 

Let wu = F(a, y) be an explicit function of the two independ- 
ent variables, x, y, and give to these variables the increments, 
Ax, Ay, whereby wu receives the increment Aw expressed by the 
equation 
au=F(a+ a0, y+ ay) — F(a,y) = F(x +ax,y) — F(2,y) 

+F(x+au,ytay)—Ma+tan,y)...(a). 
The partial derivative, or differential co-efficient, of a function 
with respect to one of the variables involved in the function, 
is that which comes from attributing an increment to that va- 
riable alone. The partial derivative, or differential co-efficient, 
of u=F(2,y), taken with respect to x, is denoted by (x,y), 
or ae In like manner, /”)(a, y), or 7 denotes the partial dif 


ferential co-efficient taken with respect to y; and F’., (a, y), or 
117 


118 DIFFERENTIAL CALCULUS. 


jae is the partial differential co-efficient taken with respect 
dady’ 


to x of the partial differential co-efficient taken with respect 
to ¥. 
Now, if 7), 7, 73, are quantities which vanish with Aq, Ay, 
then, by Art. 15, we have the following : — 
F(x+aa,y) — F(a, y) = F(a, y) sa +7, a, 
F(a+aa,y+ay)—F(a+aa, y)=F, (e@+an,y) sy + ray, 
Fi(a+ aa, y) —F, (x,y) = Fy (@, Y) 4a + 1, Aa 5 
from which last we get 
Ei(x+ aa, y) = Fy (x, y) + Fi, (a, y) da + 1; Aa. 
By substituting these values in Kq. a, it becomes 
AU ce? yyrsc+ F(x, yay tr sctryay 
+ Ff TED y) AXAY + Tr, AxAY; 


or, 


Si oe aa +- Tay +r ae + rg Ay 


wi pea tea breawsy wo 


The increment aw : a function of two independent varia- — 
‘bles is, therefore, like that of a function of a single variable, 


composed of two parts; the one, of Aw + oe Ay, of the first 


degree with respect to the increments Ax, ay, and in which 
the co-efficients of these increments do not vanish with the 
increments. The other part is made up of terms which are 
either of a higher degree than the first with respect to Ag, Ay, 
or they are terms in which the co-efficients 71, 72, 73, of the first 
powers of Ax, Ay, vanish with these increments. 3 

From what precedes, we pass by what seems to be a natu- 
ral extension of our definition, Art. 16, of the differential of a 


DIFFERENTIATION OF EXPLICIT FUNCTIONS. 119 


function of a single variable, to that of a function of two varia- 
bles. If we write du, dx, dy,.for Au, Ax, Ay, respectively, in 
Kq. 6, neglecting at the same time all the terms in the second 


member after the second term, we ae 
a 
1 4 acmemmyee 5, ae 1 mu: ly (c). 
Here du in the first ange ie the total differential of 


du du 
u, and is different from the dw in ) i, In this, as in former 


.,. du du ae 
cases of differentiation, eeay? are to be regarded as the limits 


of the ratios of the increments of the variables to the corre- 
sponding increments of the function; the distinction being, that 
now in each of these ratios the increment of the function is 
partial, and refers to the variable whose increment is the 


du du 
denominator of the ratio. We must treat ie as wholes, 


and not as fractions having dw for the numerators, and dz, dy, 
for the denominators. . It is true that du, dx, dy,in these differ- 
ential co-efficients, may be regarded as quantities rather than 
as the traces of quantities which have vanished, by assigning 
them such relative values, generally infinitely small, that their 
ratio shall always be equal to the differential co-efficients. In 


du We 
this case, =- dx would reduce to du; but this is the partial 


rae 


differential of w taken with respect to x, and should be written 


d 
du. So likewise Zy dy should be written d,w. To indicate 


du du 
that davai are partial differential co-efficients, they are some- 
du\ /du 
times enclosed in ( ); thus (z ), eS 


From Kq. ¢, we conclude that the total differential of a func- 


AO DIFFERENTIAL CALCULUS. 


tion of two independent variables is the sum of the partial 
differentials taken with respect to each of the variables sep- 
arately. 

81. To find the differential of u= F(x, y,), a function of 
the three independent variables a, y, z, denote as before, by 
1,7, 73.-., quantities that vanish with aw, Ay, Az; then 

Au= F(a@-+ av, yt ay, 2+ Az) — F(a, y, 2) 

= (a+ Aa, y,%) — F(a,y,2) + F(a+ aa, y+ ay, 2) 
—F (a+ Ax,y,2) +f (a+ aan, y+ Ay, 2+ 2) 
—F(x+au, y+tay, 2)... (d). 
But, Art. 16, 
F(a+ax,y,2) — F(a, y, 2) = F(a, y, 2) Av +7, Aa, 
F(a+Aax,y+ ay,2z)— F(x + aa, y, 2) ) 
= Fi(a-+ ac, y, 2) ay + may. 
SN Te ANSE ae =F} (x +An,y tay, %) Az + 7; Az. 
—F(x+an,y+ ay, 2) 
Also, from same article, 
Fi(x + au, y, 2) — F, (a, y, 2) = Fy (2, y, %) 4a 4 Bae; 
and therefore 
F(a + Ax, y, 2) = Fy (ax, y, 2) + Fey (2, y, 2) Ae + 1 Aw. 
So, likewise, 
Fi(t tax, ytay, 2) =F) (x4 +a, y, 2) . 
+ Ey, (2 AL, Y, BAYA MSAY, 
and 

Fi(a+taa, y, 2) = Fy (a, y, 2) + Fy, (a, y, 2 Ae+ re Ax. 
Making these substitutions in Hq.d, and denoting the co- 
efficients of the terms containing the products of Aa, Ay, Az, 
by each other, by m,, m2, m3, we have 

AUu=F (x,y, z)Ac+ Fy (a, y, z)Ay + F, (a, y, 2) dz 
trArtrnAytrAztm, ATAY+M,AXTAZ+M3AYAzB; 


DIFFERENTIATION OF EXPLICIT FUNCTIONS. 121 


or, bu ae Seay tones trae + ray + 1,42 
+m, ALAY + M,ALAZ+ MAY Az. 

From this, by the same considerations that led us to the 
expression for the total differential of a function of two inde- 
pendent variables, we conclude that 

ins oe ee a5 aay + oe de 
which may be written 
du =d,u + d,u + du. 

The course to be followed for a function of four or a greater 
number of independent variables, and the results at which we 
should arrive, are obvious. The total differential of a function 
of any number of independent variables is therefore equal to 
the sum of the partial differentials of the function taken with 
respect to each of the variables separately. 

82. In Art. 42, a rule was given for the differentiation of a 
function of an explicit function of a single variable. It is now 
proposed to treat this subject more generally. 

Let u= F'(y, z) be a function of the variables y, z, which 
are themselves functions of a third variable x, and given by 
the equations y= q(x),z=—w(x). If x be increased by ag, 
u, y, and z will take corresponding increments, which denote 
by Au, Ay, Az; then 

Au=Miy+Ay,2+Az)—F(y, 2) =F(y+ay, z) 

—F(y, 2) + Fy + Ay, 2+42)— Fly + ay, 2). 
Dividing through by 4, and in the second member multiplying 
and dividing the first two terms by ay,and the second two by Az, 


Au (y+ 4y, 2)—F'y, 2) Ay 
a AY Ax 


i et ot) UY A AS 


AZ AX 


16 


oe DIFFERENTIAL CALCULUS. 


Passing to the limit by making Aw = 0, and remembering 
that Ay and Az vanish with aa, the first member becomes ~ 
pe the first term of the second member becomes la dy . 

dx | dy dx 
see clearly what the second term of the second member be- 
comes, suppose, first, that Ay vanishes; then this term reduces 


to 


Ey, % + 42) — Fy, 2) AZ, 
y AZ Ax’ 


L(y, % + Az) + Fy, 2) 


and it is evident that, now, the factor a 


is the ratio of the increment Az to the corresponding incre- 
ment of the function: hence, at the limit, this factor becomes 
ae and the second term ge of ; and therefore we have 
ae” dz da 

du dudy | du dz 

dx” dy da ' dz da’ 


and 


alu du du 
17 dx uU zy dy + Te dz 


In general, if w= F(y,2,u,V...), Y, %, U, Ue 


functions of the same variable x, we should have 


dw dwdy , dwdz , dw du 

de — dy dah ds dein de do aa 
dw dw 
kN) BE 
dy‘ yr di 


pee aie ae 
dy) ete Prue 


tial co-efficients and partial differentials of the function w; 


- (a) 
dw = de (6). 
du 
SLADE : 
dy, hs dz, are the partial differen- 
while a and dw in the first members of these equations 
2 


are the total differential co-efficient and total differential 


FUNCTIONS OF FUNCTIONS. 123 


of the function: hence we may enunciate the following the- 
orem; viz., the differential co-efficient of a function of any 
number of variables, all of which are functions of the same in- 
dependent variable, is the algebraic sum of the results obtained 
by multiplying the partial differential co-efficient of the func- 
tion taken with respect to each dependent variable by the dif- 
ferential co-efficient of such variable taken with respect to the 
independent variable. This is the meaning of Hq. a; and 
Eq. 0 admits of a like interpretation. 

If in the function, w= ’(y, 2), we suppose, for a particular 


case, that y and z in terms of w are given by the equations 


: oe 1 a Eee and the sec- 
ond term in the second member of the equation, 

du __dudy , du dz 

ao nets’ 02.000) 


fete x; then dz— dx 


du 
dz 
of u— f(y, 2) = f(y, x) taken with respect to w. This 


would reduce to —, which is the partial differential co-efficient 


cae dus 
must be in some way distinguished from dg 1D the first mem- 
x 


ber of the equation, which is the total differential co-efficient of 
the function. This is usually done, in cases where the two 
kinds of differential co-efficients are likely to be confounded, 
by enclosing the partial differential co-efficients in a paren- 
thesis. Thus the above equation should then be written 


du /du\dy , /du 
dz \dy) dx \de) 


83. It may happen that some of the subordinate functions 
are themselves functions of the others, and thus complicate 
the example; but the principle just demonstrated is easily 


extended to such cases.. For example : — 


124 DIFFERENTIAL CALCULUS 


let b SP e, 0,2), Uf Uy es ae 
y= 9 (2); Z=w («); 

from which, by making the proper substitutions, w could be 
made an explicit function of x, and thus the differential co-effi- 
cient of w with respect to x be found. But this result may be 
reached without making these substitutions. 

Differentiating each of these equations with respect to a, 
we have 


du /du\ dy du dz du\ dv du 

to (ay) ae + as )ae* (te) ae * ate) 
dv _ /dvu\ dy dvu\ dz dvu\ 
dan a) dx hs dat Ce y, 


in which we distinguish partial from total differential co-effi- 
cients by enclosing the former in parentheses. By substituting 


dv dy 


in the first of these differential equations the values of rele yes 


= derived from the others, we get, finally, 


ul (3 “9 («) + (4 ee @) 
+(){ Gro xarro+@) (a) 


Ex. uw=yteta2y, 
¥y = COS. &, Ar, 
du Reeth 
mafia st Din Vee 3 lh} 2 
aria AL Pera + azy, 
dys ZO oe 
Ip ee. Fe 


IMPLICIT FUNCTIONS OF VARIABLES. 13s 


therefore eee = — (2y + 2”) sin. @ + (32? + 2zy) e* 
= — (2cos. x + e*) sin. w + (3e” + 2e* cos. x) e* 
— 38e** — e* (sin. x — 2cos. x) — sin. 2x; 

a result identical with that obtained by first substituting in 
u the values of y and z, and differentiating the explicit function, 
u — cos.” x + e* + e” cos. a. 

84. When the relation between the variables is expressed 
by an unresolved equation, any one of the variables may be 
assumed as a function of the others regarded as independent. 
It is often inconvenient, or even impossible, to solve the equa- 
tion with reference to the variable taken as the function, and 
thus convert it into an explicit function to which preceding 
rules for differentiation are applicable ; and hence the necessity 
for investigating special methods for the differentiation of this 
class of functions. 
~ Consider, first, a function of a single variable, which, in its 
most general form, may be written w= F(x, y)=0. LHither 
y may be taken as a function of ~, or x as a function of y. It 
generalizes our result to leave the selection of the independent 
variable undetermined. Let Aa, Ay, be the simultaneous incre- 
ments of «and y. ‘The increased variables «+ aa, y+ ay, 
are subject to the law of the function /’(x,y) = 0, and hence 
must satisfy the equation, 


F(a + sa, y+ ay) =0: 
Au= F(x + Aa,y + sy) — F(a, y) =0. 
Treating f(x -+ ax, y+ Ay) — F(a, y) as was done in the 
case of a function of two independent variables in the last 


therefore 


article, we have 
AU 0 


Se sy 


PT, T's; oe quantities that hs be Ax, AY. 


ap tmey tramway. (a); 


126 DIFFERENTIAL CALCULUS. 


Now, by whichever of the increments we divide through, 
and then pass to the limit, by making that increment zero, it is 
manifest, that since, from the mutual dependence of x and y, 
Aw and Ay become zero together, all the terms in the second 
member of the above equation will vanish except the first 
two. 

Dividing through by Aw, and passing to the limit, we have 

day dus) Vikyon dw sad 
dat dy! se de Ty dex : 


du 
dy ai 
whence ee di 
dy 
Dividing through by Ay, and passing to the limit, we get 
du 
dudx’ dw’, , dx _dy 
ay: dy dy sae hike dy du e 
dic 


In Kq. a, writing du, dx, dy, for Au, Av, Ay, and omitting | 
all the terms in the second member after the first two, it be- 


comes 


du du 
d mae rae 5 See ae, . 
Aas. dic +- a dy = 0 


85. Ifu=F(x,y,2...)=0 be a function of any number 
of variables, one among them may be taken as a function of 
all the others regarded as independent. Were the equation. 
solved with reference to the variable selected.as dependent, 
we should then have to deal with an explicit function of several 
independent variables, —a function which has no total differen- 


tial co-efficients, such as there are in the case of explicit func- 


tions of a single variable; and we are, therefore, concerned only 


IMPLICIT FUNCTIONS OF VARIABLES. 127 


with the total differentials of the function, and with its partial 
differential co-efficients of the different orders. 

Suppose z to be the dependent variable, and that the value 
of z, in terms of the other variables, is z=/(z,y...): then 


Ceri, | (Ls Yon «yee j emi 
and, considered with reference to x alone, wu is a function of a, 
and of a function of a function of z But, by the law of the 
function F(x, y,2...), w must be zero for all values of the inde- 
pendent Oe atlcs hence its partial differential co-efficients, 
taken with respect to these variables, must be zero. 


Denote by (a) the partial differential co-efficient of wu 


taken with respect to x, and, through «, with respect to z; and, 
by st a the partial differential co-efficient taken with re- 
spect to # and z separately: then, by Art. 82, 

du\ du , dudz _ 

Ge) = at a dx et 


Similarly, by adopting a like notation with reference to 


Yy,5,t..., we have 


du. du dudaz 
Gq)=qtaa=? © 
du\ du  dudz_ 0 
a Sabie ds as ee 


Kqs. a,b, c..., will give the partial differential co-efficients 
of z with respect to the variables severally. Thus, from (a), 


we have 
du du 
detie nOe dai pyre ay 
te Fa? and, from (0), Fe, ees 


128 DIFFERENTIAL CALCULUS. 


Multiplying Eqs. a, b,c... through by dz, dy, ds ... respec- 
tively, and adding the results, observing that 


du dz du dz du dz du 

da dan ds dy! + dds Ne a 
we have 

du du du du 

det bi Mahe Fd cia - + 50s e(m). 


From Eq. m, we may find the total differential of any 
one of the variables regarded as a function of all the others; 


du du du 
thus ir a ay egy age 
foarte: Roe ne AE BEETS Se 
du 
dic 
1 bp. A eS u= ay + Ba? — ah? = 0, 
Mh tion. WOE Mae 
| alice ae 
therefore cy + Bex <0. Hae. 


From the given equation, we get y= ova — 27, an ex- 


plicit function of y; and, by differentiation, we obtain directly 


dy i bx b? x 

de’) gA/ai=onige ae 
Bx. 2. u-—y> +2* — sary = 0, 

“t= 30” — Say, 7 =P — Bae, 

dy) 2%" = OY) ty ae 

des o® — aw oy? — aa’ 


a result that it would be difficult to verify, as was done in 
Lixeals 
86. When we have given the two implicit functions, 
was la. yes.) 0, uf (%,Y, See 


IMPLICIT FUNCTIONS OF VARIABLES. 129 


of the same variables, we should have at the same time 
du = 0, dv = 0, from which can be determined the differentials 
of any two of the variables considered as implicit functions 
of all the others; and, in general, if the relation between 
the m variables, xv, y, z...,18 expressed by the m equations, 
u—0,v—0,w=0..., we should have at the same time the 
m differential equations, 
itera iathoe 0 O10 ma Ost. 

and, by means of these, could determine the differentials of m 
variables regarded as functions of all the others. 

If the number of variables exceeds only by 1 the number 
of equations expressing the relations between them, one of . 
the variables alone can be independent; and we may find the 
differential co-efficients of all the others regarded as functions 
of this single variable. 

Let us have n equations, 

tigteea hy (Opie Boa 6) a= 0; 

Merny (an Yeti « bjoa— Oy 

it eect Liye (2D) ned. «== 0, 
between the n + 1 variables a, y,z...¢. 

Differentiating all of these equations with respect to x 
taken as the independent variable, we have 


Cee OU, dy. du; dz du, dt 


eed) de de da at de” 
ewe. dy du, dz du, dt 
eee Gs de haa 


Gee au oy du, dz du, dé 
Pd. dak? neds. dx 


There are n of these differential equations involving the n 
17 : 


130 DIFFERENTIAL CALCULUS. 


required quantities, dy be i: ag which may therefore be 


dx’ dx da’ 


determined. 


87%. When the variables enter the function in certain 
combinations, the results of differentiation take special forms, 
and peculiar relations exist between the partial differential 
co-efficients, depending on the manner in which the variables 
are combined. We shall first consider the case of homogene- 
ous functions. A function is said to be homogeneous when 
all the terms entering under the functional symbol are of the 
same degree with reference to the variables. Thus 

E(x, y, %) = ax® + by? + ca? + 2eyz 
is a homogeneous function of 2 dimensions, and 


Y 


is a homogeneous function of 0 dimensions. A property of 
such function is, that, if all the variables are multiplied by 
the same quantity, we obtain for the result the original 
function multiplied by this quantity raised to a’ power whose 
exponent is the number denoting the dimensions of the 
function. Therefore, if F(x, y,z...) is a homogeneous func- 
tion of a dimensions, and ¢ denotes a new and independent vari- 


able, we have 
EE tec, ty tz,.3. 1) = EO 
Put $0. Ub, ty UB; te 
EU, 0, .. 3) ECG oy 2 eee 
and differentiate both members of this equation with respect 


tot: the result is, 


dFdu dF dv . dF dw 


——— —— = — eee taht — (Tf oe : 
du dt cae PEED Be dt zt ING, Y, 2 ve a)e 


IMPLICIT FUNCTIONS OF VARIABLES. T3d 


But a as Oe 


therefore 
dF dk ak 
* a dy dw 
Now, since ¢ is entirely arbitrary, make ¢ =1; then 


Cie onlin Glen lug) ex 
du dx’ dv dy * 


pee arige CL Ys Riess c)s 


Mae 7,20 — %..,..,, and 


whence we have 


cet lag tae ee Se ey pares). 

The first member of this equation is the sum of the. products 
obtained by multiplying the partial differential co-efficients of 
the function, each by the variable to which it relates; and the 
second member is the primitive function multiplied by the 
number denoting the degree of the function. 

If the function is of 0 degree, 


dF dF dF 
Be dy ee da te = eT 
Ex. I. F(a,y,2) = ax? + by?’ + cz? + 2eyz 4+ Afzx + 2gay, 


ae Zax + 2fz + 2qy, re 2by + 2¢e2 + 29x, 


dix dy 

dF _ 2cz + ey + 2fa, 
and a = 2: therefore 
(2ax + 2fz + 2gy)a , 
+ (2by + 2e2 + 29x) y + = 2(ax? + by? + cz? 4+ Qeyz + Bea 
+ (2ca + 2ey + 2far)z + 2gxy), 


an identical equation. 


132 DIFFERENTIAL CALCULUS. 


edi Titre 
exe ay LG ee? dey’ dy y? 
a= 0: therefore 
dk LE 2) Sy: <2 ae 
ae dy yy yy ‘ 
88. Let us next take the case of the function of the alge- 


braic sum of several variables, x,y,z... If the function be 
u=F(eatkytz+---)) and we put ctytet..-=#, 
it becomes wu = F(t). 

Now, if the original function be differentiated with respect 
to x, y,%... separately, we shall have, by reason of the equa- © 
tion u = F(t), 

dF edFdi dF _ dF dt dF dFdt 
dx dt dx’ dy” dt dy’ dz” dt dz 


But the equation x t+ytzt---=¢ gives 
dt dt dt 
a] aS te ee tS SS +6 apt 
da ! dy a dz = 
therefore 
dF dF dF 
dg ay 7 dee 


that is, the partial differential co-efficients of the function are 
numerically equal. 


du du 
x. lo we (oy Tian a i na + y)r-}, 
Db. Te (x—y)", oH — =e 
du du __ sf 


Hox, '°3, u=IV a ty, da dy 2a ty) 


du du 1 
Ex. 4. uxlvxz—y, da dy a—yy 


SECTION IX. 


SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE IN- 
DEPENDENT VARIABLES, AND OF IMPLICIT FUNCTIONS. 


89. In Sect. IV., rules were investigated for the succes- 
sive differentiation of explicit functions of a single variable. 
We now pass to the successive differentiation of functions of 
many variables, all of which, at first, will be supposed inde- 
pendent of each other. 

By Art. 81, the total differential of a. function of several 
variables is the algebraic sum of its partial differentials; and 
it is evident that each partial differential co-efficient is, in 
general, a function of these variables, which may be again 
differentiated with respect to a part of the variables, or with 
respect to the whole of them. These operations give rise to 
what are called partial and total differentials, and differential 
co-efficients of the different orders. 


90. lfu=F (a, y, 2...) be a function of the independent 
variables x, y,z..., then du, d’u, d*u...d"u..., standing 
by themselves, will denote the first, second, third... total 


differentials of the function. = is the first partial differential 
wx 


co-efficient of uw taken with respect to w; and is OG, Ol atk: 


is the corresponding partial differential. us e pee 188 is 


dx\dx/ dx? 


the second partial differential co-efficient with respect to x; 
138 


134 DIFFERENTIAL CALCULUS. 


2 
and the differential corresponding to it is st da? OTe 
x 
2 
ay at pees is the second partial differential co-efficient, 
dy\dx/  dydx 
taken, first with respect to x, and then with respect to ¥; 


dydx, or d,d,u. 


2 
and the differential answering to it is oo 


dydx 


3 
a Tt) a a ef is the third partial differential co-efficient 
dy \dx’ dydx” 


of the function obtained by differentiating twice with respect 


to a, and then once with respect to y; and to this we have 
on yen ib ‘ 2 E 
the corresponding differential ARE dyda*, or di.d,u, which 
may also be denoted by d%:,u. In like manner, the notations 
dtu d‘u hh 
a : ld indicat 
datdyde’ da*dyde dx*dydz, d®.d,d,u, di2,,u, would indicate 
four differentiations: one with respect to z, one with respect to 
y, and two with respect tox. From what precedes, the significa- 
QUMtnr+P---y UMrr+P---.y 
dx™dy"dz?...’ damdy” dz... 
dd" d...u, dm? u, will be readily understood. 


ane ee 


tion of the notations da™ dy" dz? , 


The remarks in Art. 81, in reference to partial differential 
co-efficients of the first order, are equally applicable to those 
of the higher orders. Keeping in view the principles there 
laid down, there will be no risk of confounding any order of 
partial differential of the function with the total differential 


of the same order. Thus, in oe the d?u is always. associ- 


dady’ 
ated with dxdy written below ae and in this way the con- 
struction of the expression indicates both the character of 
the differential co-efficient, and the variables with reference 
to which it is taken. 


It is often convenient to attach to the symbol of the func- 


FUNCTIONS OF TWO OR MORE VARIABLES. 135 


_ tion the characters by which are denoted the order of differ- 
entiation, and the variables involved in the operation. Thus, 
Fy (%,y,%---), Bry (2, y, +++), Ha, (Gy, %-.-), have re- 
tively the same significance as atlas Les as 
Ree 5 dx’ dxdy’ da*dy’ 


above explained. 

91. -Before proceeding farther, we must prove, that, in 
whatever order in respect to the variables the differentiation 
of a function of many independent variables is effected, the 
result is always the same: that is, if w= F(a, y,z...) 1s to 
be differentiated m times with respect to x, and times with 
respect to y, the result is the same, whether we perform the 
m «-differentiations, and then the n y-differentiations, or re- 
verse the order of differentiation in respect to w and y; or 
perform first a part of the m z-differentiations, then a part 
of the » y-differentiations; and so on until the whole of the 
m and n differentiations are effected. 

This principle may be demonstrated as follows: Take the 
function w= f(x, y,%...) of the independent variables 
©, Y,%... Suppose, in the first instance, x to be variable, and 
all the other variables constant, give x the increment h, and 
develop by the formula of Art. 61; then, in the result, suppose 
y to be variable, and all the other variables, including z, to be 
constant, give y the increment k, and develop the terms by 
the same formula. The final result will be the same as that 
we should have reached by giving the increments to a and y 
simultaneously. 

Changing x into 2 + h, then, Art. 61, 


DGG yeaa.) d! (00,Y, @..<-) + hl, (2, y, 2...) 
ie 
+5 Fa (e+ ah, RCAS Pa ea | (1); 


136 DIFFERENTIAL CALCULUS. 


in which #';(@-+ 0h, y,2...) isa function 2, h,y,z..., which 
remains finite when h = 0. 
If in (1) we change y into y +4, the first member becomes 
F(e«thy+hk,z...); 
and the terms in the second member become respectively 
Fic,ythkh,z...)=F (a, y,2...) + ki Gaps) 


hae a 
+ a Lie (4 Y + 92k, &---), 


AF, (c,y kh, 4...) SAE (2, y, 2.. +) ani eee 
hk? mr 
ap SO TTES (v, y+ Osh, 2...) 


2 
F'(@+hythz...)=” = FY, (@ + 1h, Y, a) 


Ra ip 
+= Fi, (©+ 6h, y+ Ok, 2...). 


Making these substitutions in Eq. 1, we find 
(0, Y, & 0s.) PRET yy aes) 
+ hE) (x, y, 2...) 
| a +AKE,, (x, Y,%.. +) 
+ 7 Fa (@ + Oh, Y &-:) 


Bahl 1h) Key ata eae 


hk? "I 


we ee (x +6, h, Pniey ire F 


If we begin by giving y its increment, we shall have the 
equation 
Pay +kh2...) =F (x,y, 2...) + kF, (x, ys eee) 
k? a 
hy Fy (Usa, 2 ey 
and in this, giving to # its increment h, and developing the 
terms as was done above, we have 


FUNCTIONS OF TWO-OR MORE VARIABLES. 137 


[ (a, y, 4...) AP, (x, y, %---) 
+E, (a, y, 2. +) 
AE js (By Beek ke) 


$5 P(e + Oh, Y, %+-) 
kee 2 
BFiaxth y+k,2z...)= + 5 Fe (2, y + Oh, 2...) 


kh? mt / 
+-5 Fya(e+o TRU icra) 


4+ Be Fyn, (@+ Oh, y + Oh, %...). 
It is to be observed, that, in the several preceding equa- 
tions, the factors Ff), (x + 6,h, y,%..-), Fia(®, y+ 9:k, 2°..), 
&c., of terms in the second members, remain finite when / and 
k, separately or together, become zero. 
Equating these two values of F(a+h,y +h, z...), sup- 
pressing terms common to the two members of the resulting 
equation, and dividing through by Ak, we have 


wr k yr 
FY (0, 4,2. +E P(e y+Okye...) | 


i we 
fey 30 (2+ Oh, y + Ak, oH 


ai h rr 
jig (2, Y, 2) Fee hs (w+ Wh, y,%...) 


a k we 
| +5 Fi (@+ Ov hy + Ob, 2...) 


This equation must be true, whatever the values of / and k. 
Make h=0,k = 0; then , 
Frey (%) Y, % +++) = Eye (®Y, #) (2). 
The first member of this equation is the second partial dif 
ferential co-efficient of the function obtained by differenti- 
ating, first with respect to x, and then with respect to y; the 


second member is the second partial differential co-efficient 
18 


138 DIFFERENTIAL CALCULUS. 


which comes from differentiating, first with respect to y, and 
then with respect to x It is therefore immaterial in what 
order the differentiations are performed. 

This theorem being demonstrated for derivatives and dif- 
ferentials of the second order, it can easily be extended to ) 


those of any order. Suppose we start with Whether 


z 
this be differentiated, first with respect to x, and then with 


respect to y, or we invert the order of differentiation, the re- 
sult is the same by what has been proved. So that 


dui d*u 
dadydz ~ dydadz 
But the order of differentiation with respect to z, and either 


x or y, may also be inverted ; and therefore 
CLP Us TN LA ed ee 


dxdydz dydadz~ dadzdy 


and generally, for the function v= F(a,y,z...), 


(me rnr+p U Ar tmtrPy qdm@rp+n U 
da™dy"da? — dy*da™dz? — da™dz?dy” 
re 
Exe]: pray AI & 
du any du 4x y 
dx (a? + y?)? dy (oa 


au  _8ay(x?—y?) du. sage) 
dyda  (x* + y?)) " dady (a? + y?)F 


Ex. 2. tan abe; 
EER AG) Ce! Tee 
dx ay?’ dy wy’ 
azu ih a? — y? REY. v2 —y? 


dyda (x+y?) dady~ (Fy? 


, 
FUNCTIONS OF TWO OR MOR [7 ABET 139 


a able: is zero. Now, since the he len Sea\ 
function of the Slama Ea ealee ge Gigs réspeet to 
vat end the corre- yf 


co-efficient by the differential of the variable to which it re- 
lates, it follows that, in subjecting such differential of the 
function to further differentiation, we may set aside the differ- 
ential of the variable as a constant factor, and operate on the 
differential co-efficient alone ; restoring in our final result the 
constant factors set aside: thus, if w= f(x, y,z), in which 
x,y, and z are independent, then 


etal, (ant, Bae == a dx, 


d,d,u = dad, F(x, y, 2) = da Fy, (x, y, z)dy 


gene Lady a dxdy, 


Mu 
dady 
dd ,d,d,u = daxdyd, Fy, (x,y, %) = dadyF')’ (x,y, )dz 


deu 
ey mas, ONT he ea 
pe Y; 2)dady di dadydz dadydz, 
and, generally, 
a2 dy Uy te = Fenn sm (0, Yy 2) ce dy de? 
QUMrrr+Py 
m nN Any DP 

= quam dy"da da™dy"dz?. 


93. By Art. 81, the first total differential of the function 
u = F(a, y,2), of the variables a, y, z, is 


du =F de + oi y soil seks (eer 


Taking the total eer pa each of the partial differential 


du du du 
co-efficients rE Nae 7,7 We have 


140 DIFFERENTIAL CALCULUS. 


de = mete Ut oak 
Ca) ats ap 
204 
4G) aa? eg te 
and therefore 
fo ae dat 42 2 tag 


+ gi 7, dad _ a 4, te (2) 


Proceeding with (2) in the same manner nee we did with 
(1), we should get the third total differential of the function ; 
and so on. 

For the function w= F'(a,y) of the two independent vari- 
ables a and y, the successive total differentials will be 


et ge 


d?u 
2 
dt = 5 de” pee i edy + Gat - 
au fee ad*u 
au = x? d Ba 5 a: dy? 
nda sh y ail cay 
au .,. 
Bae i 
du AMA SY, 
lw = ——~ dy” Outs Vek pies, 
VO Ten Br _ 
n(n—1) d”u da”—*qy? + tee 


tS 1.2. da" —*0y? 


nm(nm—1)... (n —(n —2)) d™u 
ss 1.2..-(n—1) anag 


FUNCTIONS OF FUNCTIONS. 141 


the law of the co-efficients being the same as that in the de- 
velopment of (1 + x)”. | 
dix. 1, Meco e 
du = yzdx +- xady + xydz, 
d?u = 2 (adydz-- ydadz + zdady), 
d?u = 6dadydz. 
Ex. 2. w= (a? + y?)?, 


mea end Se Te OE Ova BUA 
: da (w+ty?)? dix? (x? + y2)3 dic3 (x? + y2)8 

du __ y Oe ee ge ant ey ire 

dy (e+yrft dy? (oye dy (wt yt 
meer Ce Byler ye) 


dy (a? +y) dutdy (a? +y?)! 
du ¢(2y? — a”). 
dady? (a? + y?)? 


1ST own, (~ 3xy? de® + 3y (2x? — y?) du*dy 


+ 3a (2y? — a”) dady? — Byotdy) Bee. 


(+yi 
Ex. 3 ip ee 

Ee ed Tpke ght S ti gd erate 

du — bet + by dl? — brewtly 

dy ” dy? 
d*u 
Sees ax-+by. 
a. abe : 

Mu = (a*dx? + 2abdxdy + b? dy?) erty 
= (adx + bdy)e +, 


94. If, in the function s= Fu, v, w), u, v, and ware 
functions of the independent variables wx, y, and z, we have a 


142 DIFFERENTIAL CALCULUS. 


case of a function of functions of independent variables ; and 


the first total differential of s is 
ds ds ds _ 
—e —d 4 ty, : 
ds FE ate ves z (1) 


But since u, v, and w are all functions of a, y, and z, the par- 
tial differential of s, regarded as a function of a, is (Art. 82), 


ds ds du ds dv ds dw 
deo” du dete} do ee 
ds ds du ds dv ds dw 
] dy = — —dy+—— — — dy; 
Be ay! andy 2) du dae ar 
ds ds du ds dv ds dw 
aes ayes — — daz, 
eae ign dude nde 
The total differential of u is 
du 


du = — de r+ yt eS 


and, for the total differentials of v and w, we have like expres- 


ds ds 
sions: therefore, by substituting these values of — cE dx, ay dy, 
Y 
d. 
=, de in (1), and uniting terms, we have 
ds 
th cP du ae , +4 Tae dw. 


A second ait Ae ee give 


d’s d’s ds 
2 of——— 2 w 
Gist me ie sa dv Sale! ~ dw +24 duh dudv 


dudw + 2 abe dudw He — ae 


dvudw 


2 
Br ae 
bak as ds 4. ; 
ie ao” 
and from this we pass to d’s, and so on. 


The general rule is, then, to differentiate as if uw, v, w, were 
independent variables, and substitute in the results the values’ 


HOMOGENEOUS FUNCTIONS. 143. 


of du, dv, dw; d?u, d?v, d?w..., derived from the equations 

giving w, v, w, in terms of the independent variables a, y, z. 
If uae + by + ca +d, v=a’et by +c2+d’, 

wale + b/y + 0/% + d”, 

are the expressions for wv, v,w, in terms of a, y, z, these func- 

tions of the first degree, with respect to the independent vari- 

ables, are said to be linear. In this case, we should have 

eee 0, 0) de = 0; dee. s+ 

and the successive differentials of s = F'(u,v,w) would then 

have the form of the successive differentials of a function of 

three independent variables: thus 


ds d’s ds d*s 
Ligh. 2 2 2 
d's— — du sre ra + a, dw aude" 7 dudv 


ey ae dudw + 2 leah dudw. 


Ex. 1. s=fF (u,v), u=—axc+by+e, v= ale-Lbly +e, 


n __ a*s ds n—l d's vy", 
eee ae du” ae — iq du du-+.. a dU". 


Ex. 2. s=f'(u)F(v), u=ax+ by+c, v=a'a + byte’, 
ds = EF" (u) F'(v) du + Fu) £” (v) dy, 
C3 ae E’(v) du® 4+- 2F"(u) “ v) dudv + a ee ee 


de = Fw (u) Flo) oe Le nPo- D(u) Fi(e) oie ee 
+ nf" (u) F°—” (v) dudu”—! + F(w) ae ee 
95. If the function s = F(a, y,z) of the independent varia- 
bles, x, y, 2,18 homogeneous, and of a dimensions, then, by Art. 


87, 
dF dF. dF 


da Yay tae 


It may be shown that similar relations exist between the func- 


Peat ao? ) = 


tion and its differential co-efficients of the higher orders. 


144 DIFFERENTIAL CALCULUS. 


Since the function is homogeneous, if we change a, y, and z 
into tx, ty, tz, and make u = tz, v = ty, w = tz, we have 


Pu, 0; w) = t F(x, 9,2). 


Differentiating this twice with respect to ¢, observing in the 
d /du\ d /dvw\ d /dw 
second differentiation that — ai ( i) sila aia) are each 


zero, we find, 


dF du .dFdv . dF dw 
du dt ' dv dt ' dw dt 
CKdu? PE dv  dFdw* | 
au? at S det dS dw dt 
d?F’ du dv d?F du dw 
eS nog By =< Ley | §% ‘ 
ste dudv dé dé dudw dt dt | a(a—-1) 7 F(a, y,2) 


ae at?) F(a, Y; a). 


, a?F dv dw 
AsO aon 


But Sy = 2, ate Y; e — 2; and, if¢=1, the second partial 
differential co-efficients of the function with respect to u, v, w, 
become the second partial differential co-efficients with respect 
to x, y, %, respectively ; and hence the last of the above equa- 
tions becomes 

2H 2 2 
PE yi 0 


ad? ar a’ 
Dine pe eh 
Te pene dady ady potas dadz Tea dydz 


= a(a—1)t**F (x,y,z). 


By a third differentiation, we should get 
a 


pst Wie 
let ain of ha: 
DPF nis = a(a-1)(a—2)t? F(x, y, 2). 
+32? Y itayaae xy Ga iapas 


DIFFERENTIATION OF IMPLICIT FUNCTIONS. 145 


Example. F(x, y, 2) = ax? + by? + cz? + 2ery + 2fxz + 2gyz, 
3 eae: iG ar 


"pn Ga = Ga = % 
ar fa ak 
Og va ay ee 
dady ” dads 1 dydz sath 
oe F dar ae be pace F 


y 
eee ae gst. aed 
ak d?F 
2 Bie iy eae 
ene dada ged? dydz 
2 (aa? + by? + cz? + eay + 2fxa + 2gyz) = 2K (a, y, 2). 


96. To express the successive differential co-efficients of 


implicit functions, take the function wu = F(a, y) = 0, in which 
y is implicitly a function of w; then, by Art. 84, 

du, du dy _ 

iad dy da 


The first member i. i is another function of w and y, which 


=—0 (1). 


denote by v; whence v=0. Differentiating v= 0 as we did 
u — 0, we have 
dv dy _, : 
qe ASE dy da Bee) 


| 2 2 
oa dv du, du dy , dudy 


dx dx? ' dady dx ' dy dx?’ 
dv du , du dy 
me dy dady ' dy? du 
These values of on 7 substituted in (2), give 


os 9 Wu dy d?u (dy du d?y _ 
dni’ ” dedy dot dy? = (ae) as): 
2 
From (1) and (3), we find the values of gh ath Kqs. 1 and 


3 are called differential or derived equations of the first and 


second orders respectively; and, with reference to them, 
u — F(x, y) = 90 is the primitive equation. 
19 


146 DIFFERENTIAL. CALCULUS. 


: | at od 
The above process is somewhat simplified by putting ages 


then 


du | du 
PM py eng LAE 
aa ae (1’) 
dv. dtu au du (dp 
aE ese : dz da? us dady © Ay alae 
dv .d'u du du /dp 
ot dy dady + ap? © wa) 


These values in (2) give 


du du (dp du dite ip eae 
(2 8(6) Ze ee 


du du fy dp\ | - 
det dady? + Gp? to 


But (2 ) me = te (Art. 82); and hence 


au au du, , du dp 
“he — 1p? + 1 = OF 
- 2 Rea de dy da (9), 


d d. du ad? 
2 d’u dy Y\ at oe 
ce Te eal dady et ae dy’ ag | T dy da? 


d. 
We call attention to the notations Ge ee ee by re- 
marking that p is generally a function of «and y; and that 


dx/’ \d 
tion, the first with respect to xv, and the second with respect 
to y: whereas ed is the differential co-efficient of » with re- 


spect to x, p being considered, as it is, a function both of « 


G ) & ) are the partial differential co-efficients of this func- 


and of a function of a function of x. Thus suppose, that, by 
solving the primitive equation (x,y) = 0, we find y= f(z), 
then 


DIFFERENTIATION OF IMPLICIT FUNCTIONS. 147 


d 
p= (2), =f"(2). 
Suppose also, that, without solving the primitive equation, we 


find p=9(2,¥) =9(2,f(a)); then 


a ee 
(=o ens (F) =a au). 
But, by Art. 82, 


d / ' dl 1 
= 92(% Y) +99 (2, Y) = S'(a):.. Q). 


This points out the necessity of distinguishing, in certain 
cases, partial differential co-efficients, such as those of p in 
Kq. a, by the parenthesis, or some other mark, that, in 


the course of an investigation, they may not be mistaken for 


others, as 2 in Kq. 0, of the same form, but having a differ- 


ent significance. 


The value of oy deduced from Eqs. 1 and 8, or from I’ 


da? 
and 3’, 1s 
d?u /du? d*u. du du. d*u /du2 
dy at je) * dxdy dx dy ' dy? a ; 
TE ah du\3 
Mi) 


The expressions for the higher orders of differential co-effi- 
cients of implicit functions are so complicated, and so little 
used, that it is unnecessary to proceed farther with this divi- | 
sion of the subject; but we will conclude it by giving the 
differential equation of the third order of the implicit func- 
tion y of the variable x, given by the equation wu = F(x, y) = 0. 


This differential equation is 


148 DIFFERENTIAL CALCULUS. 


d*u d*u dy d'u /dy\? d'u/dy 
dix? ide dx*dy dx eS dacdy’ cs 195 dy? ete 
d’y , dudy\dy' du dy _ 
3 ed Lis 
a (ser dy? es dx? ' dy dx’ Sy 


97. Suppose we have given the two simultaneous equa- 


tions 


ie shi Ge a! Pt) =m) tall b bp 

U = fat, 20 ae 
It is theoretically possible, by combining these equations, to 
eliminate either variable, and get an equation expressing the 
relation between the other two from which the successive dif 
ferential co-efficients of one of these regarded as a function 
of the other might be obtained. But without effecting this 
elimination, not always practicable, we may proceed as fol- 
lows : — 

Suppose «x to be the independent variable, and differentiate 

(1) with Ne to x; then (Art. 85) 

du dy , dudz 


wt 95 de ae ae = Oe 
In like manner, from (2), 
du dy , duda _ 
a Pal dy dx ' dz dz Se: 
From (3) and (4), we find 
du dv __ dv du 


dx  dudv dvdu 


dy dz dy dz 

and 
dv du du dv 
dz da dy dx dy 


Be Sy 
ax dv dus du dv 


DIFFERENTIATION OF IMPLICIT FUNCTIONS. 149 


The first members of (3) and (4) are functions of a, y, 2; and, 
by differentiating them with ae to x, we have 


d*u d’u dy d’u dz = Ge d’u dy dz 


dx? 1 dady dx ee? dadz da +o dy? vox dydz dx dx 


dz du d*y dud?z i 
+o Ae ae apa dada 7 ee 


and 


dv 9 d°v dy dy dz aaa ay dy dz 


dic? dicdy aes dade EQh oie da dydz dx da 
d’ Gal du d*y | dvudz 


Pb a Fare, enh See) FA 

nae Ban rea: es tla Oe 

From (7) and (8), by substituting in them the values of 

BS ee in (5) and (6), we may deduce the values of cd 
ie 


and oe They may also be found directly by differentiating 
(5) and (6). 

98. For an application of the methods of successive differen- 
tiation, suppose we have the single relation wu = F(a, y,z) = 0 
between the three variables x, y,z; then z may be consid- 
ered as an implicit function of the two independent variables 
a. 

It is required to find the first and second orders of the par- 
tial differential co-efficients of z with respect to # and y without 
solving the equation u= F(a, y,z) = 0. 

The first partial derived equation with respect to @ is 
(Art. 85), 


du dz 
Bate Bis teen 
and that with vt to y 18 
du dz 


150 DIFFERENTIAL CALCULUS. 


du du du 
dx’ dy’ dz’ 
of uw, taken on the supposition that the variable a, y, or z, to 


in which are the partial differential co-efficients 


which they separately relate, alone varies. 
Kqs. 1 and 2 will give oe 7 Differentiating (1) with re- 
spect to x, and (2) with respect to y, and also either (1) with 


respect to y, or (2) with respect to x, we get 


du di aN ia fda\t s duad2 
ee ? Uega.dig's dat i Th a5 ape le (3), 
du Mu dz dz\2 du dz 
dy? it? dydz at ast sen da ia (4), 


du Ouida: d'u-dz . d’u de dzaaiaiuaeee 
dxdy ' dedx dy 7 didy da T aa dy di T a dady 
and from the five Eqs. 1, 2, 8, 4, and 5, we can deduce 
Bemeeiee 02" On mies 
da’ dy’ da®’ dy?’ dzdy 

d*y 


Ex. 1. Given y* + x«* — 3axy = 0, to find the value of meee 


=0 (5); 


The first differential equation is 


and the second, 
a? 
(y? — ax) (i) 2a e+ ay (OE) +2x=0 (2). 


Substituting in (2) the value of taken from (1), we find, 


after a little reduction, 


2 
(y* — ax)? a — 2a(ay — a*)(y*? — ax) + 2y(ay — x*)P 


+ 2x(y? — ax)? =0: 
whence 
ay __ 2a(ay — x?) (y? — ax) — 2y(ay — x")? — 2a(y? — ax)? 
dz? (y? — ax)? 72 


DIFFERENTIATION OF IMPLICIT FUNCTIONS. 151 


and this, after performing the operations indicated in the nu- 
merator of the second member, and reducing by the given 
equation, becomes 


dty - da* xy 


Oe (yy? — aa)" 

Ex. 2. Given 6?c?a? 4+ a?@y? + a’?b?2? — a’?b?c? =0, to find 
I Sell SR ae 
da” dy” dady’ 
az c?(a?2? +c?x7) dz c?(b?2? + c*y”) 
i an aiet ae ae bt? . 
ieee Cry 
eee oe 0" 2" 


SECTION X. 


INVESTIGATION OF THE TRUE VALUE OF EXPRESSIONS WHICH PRE- 
SENT THEMSELVES UNDER FORMS OF INDETERMINATION. 


99. Ir sometimes happens that the expressions under con- 


sideration assume, for particular values of the variable or 


. . > 0 
variables involved, some one of the forms ~, +2, 0x0, 
ie @) 


| 0 
0°, 0°, + 1°, o —o, called forms of indetermination, though 
he value of the expressions may be determinate. Our object 
now is to establish the rules by which may be found the true 


value of an expression which reduces to any one of these forms. 


100. Of the Form - This form can only result from a 


fraction in the numerator and denominator of which there is 
a common factor, which factor becomes zero for the particular 
values of the variable or variables which reduce the expression 
P (x —a)”™ 

Q («@—a)”’ 


may or may not be functions of «; but, if they are, they do 


to 4 -Thus take the fraction in which P and Q 


not contain the factor « —a, and therefore do not become zero 
when «=a. If in this fraction, as it stands, we make « =a, 
it takes the form . but if, before giving «x this value, the 


t 


Hi sae 
fraction be written — (#—a)”~", 1t1s seen that the true value ~ 


of the fraction for c=a is 0 if m >n,o if m < n, and A 


if m=n. This suggests the following rule for the evaluation 
152 


INDETERMINATE FORMS. 153 


of expressions which take this form; viz., discover, if possible, 
the factors common to the numerator and denominator of the 
fraction, and divide them out. What the result reduces to by 
giving the variables their assigned values is the true value 
of the expression. 
vetoar—3e—3 0 
= - Btens he cane Geo Ee 
ie ear e g When # /'3 
x + x? — 84 — 3 ee er) 2 te 
e* — 20? —34+6 (w?—3)(a—2) aw—2 
1473 
— for «= V3. 
Js, ee 


Many cases of the form : may be treated as follows :— 


Example. 


but 


i 
Take the fraction Pre ein which becomes D when «=a. 
ah? 0 


(a? — a")! 
Make «x =a-+h; then 
Sense 7): a e ace i = =.0. when 
(a? + 2ah+h?—a*)s h3s(Qa+h)s (2a+h)s 
h = 0, which corresponds to # = a. 
oe 0 
Also the fraction Ve— Va is Vea O = — for g — a; mak- 
Ja — a iG 
ing «—a-+h, 
—Wath—Va+vh _ Vath —(va—Vh) 
/ 2ah bh +h? / Qah + h? 


Multiplying numerator and denominator of this by Va+th 


+ (Va —wWh), we find 
2*/ah 1 


(Qah + 12)! (a+ ht + (Wa —Vh)) ~ / 2a" 


for h = 0, after dividing out the common factor h?. 


The examples already given have been solved by common 


algebraic transformations; but most of the cases which present 
20 


154 - DIFFERENTIAL CALCULUS. 


themselves can be more easily solved by means of the differ- 


ential calculus. 


101. Suppose the fraction to be Fe aot and that both F(a) 
and f(x), as also their successive differential co-efficients up 
to the (n— 1) order inclusively, vanish for «=a; then it 
has been proved (Art. 56) that 

Fiat hy Pease on ie 

Path) f(a oh)” 

and consequently, by making 2 = 0, we have 
Ba) _ F(a) 
AY GSRE@S 


Hence, to obtain the true value of the vanishing fraction 


Hm) when « =a, form the successive differential co-efficients 


J(&) 


of both terms of the given fraction until one is found, whether 
of numerator or denominator, that does not vanish for z=a; 
and take the value, when «=a, of the fraction whose terms 
are respectively the differential co-efficients, of the order of 
that thus found, of the corresponding terms of the given 
fraction. 

If one of these differential co-efficients vanishes, the value 
of the fraction will be 0 or «, according as it is that of the 
numerator or of the denominator; and it will be finite if the 
first of the differential co-efficients that do not vanish is of 
the same order in the two terms of the fraction. 

Tix ; OT rue 2 a). 

sin. & 0) 
d(x) = e* —e-*, P(x) = e* + e-*, /(¢) ae 
Bate ee Ce 


Gann Cray oe aa 


% 


INDETERMINATE FORMS. BS 


Ex. 2. 2 — sin. x its b.— cos. 2 an sin. a 
aie tion? oe eae Ore fin 


Ex. 3. ( Ue ) =(5) nih 
e—1/,—1 LY 1 


102. Form — If the two functions /'(x), f(x), become 
£2) 


infinite for « = a, the fraction Fiz) reduces to - But io 


ii 


this case the fraction may be put under the form LE), which, 
F(a) 
for x = a, becomes zs and may therefore be treated by the 


preceding rule. ‘Thus 


: f(a) . 
Fa) f(a) (4)) ~  (F@) pray | 


Meo Ka ( fla)) 7). 
@ (roy | 
whence ah — 7 mi and the true value of the ratio aa = 


F(a) 
J’ (4) 


Tf all the differential co-efficients of both terms of the fraction 


is the value of 


become infinite up to the (7 —1)™ order inclusively, then 
Bea) Se F(a)’ 

and the true value of a ratio, that, for a particular value of the 

variable, takes the form =, is the value of the ratio of the dif 

ferential co-efficients of the order of that first found, whether 


156 DIFFERENTIAL CALCULUS. 


of numerator or denominator, which does not become infinite 
for the assigned value of the variable. 
Example. For «= 0, 
l 
l = . 9 e 
ee O-: sm? ae. 2 6in, cede 
cosec.x 0 wcos.v cos.e—a@sin.x 


103. The rules which have been given for finding the true 
value of ratios which take the form ; or — are applicable for 


infinite as well as for finite values of the variable. This fol- 
lows from the fact, that the reasoning by which these rules 
were established requires only that the value attributed to a, 
causing the fraction to assume the one or the other of the above 
forms, should be the same in both terms, but does not involve 
any supposition in regard to the magnitude of this value. The 
rule depending on differentiation may be demonstrated directly 
when the form of indetermination comes from the hypothesis 
ee 

Represent the terms of the fraction by /'(x), /(x), as before, 
and suppose, that, for « = 0, we have either (x) =0, f(x) =0, 


OF 2? (2 )— 00, () = cor; tthen; putting | for &, 
F(a) _* () scales a ee 
BQ" TOA 


but, by rules already given, 


[70)| _[s"Q)] 


= 


INDETERMINATE FORMS. 157 


hence 


ae f 
F(2) _|73)| 2 | 2 (ae 3 
Oe || orf ee 


Ex. 1. For x= we have, whena > 1, 


a*_atla_ 

Tig as Wine 
ee ly Len 7. —"oo.., 

fe 1 _ 9 

x ala 


Ex. 3. Whenw=—o, and n is the integer which immedi 
ately follows a, 
a* _a(a—1) (a 2)...(@ —n+1)_ 4 
oF erage 
104. Form 0x. Let F(x), f(x), be two functions of a, 
one of which becomes 0, and the other infinity, for a particular 


value attributed to x. Forx=a, suppose f(x) =0, f(x) =a. 
The product may be put under the forms u= F(x) x f(x) = 


ae Ae as ; the last two of which, for the assigned value 
F(z) (2) ‘ 
of x, take respectively the forms 0” =, and can therefore be 


treated by the preceding rules. 


Exel, w= 1(2—5) x tan, 57 = OXen when «=a. 
a 


2a 
But 1(2 )xt PER Reset? a 
2 1 1X 
Gots == 
tan. 2 ‘3 


158 DIFFERENTIAL. CALCULUS. 


Geo 
(2-2) pple 
and - eles i _4 
ote ’ ti J 5 
[ Ue elo | ET: a2 = 
[ 2a. rt=a 
ie oe or 
cla = —0XKo, 


lar las aie: 
aclac = eth and eae = ee 0. 


Ex. 3. «™”(lxv)"=0xXo for «=0, when the exponents m 


and are positive. 


Make «= - then 2” (Ie)\"—(— 1). This, by? Bxo3;* 


oe 
Art. 103, is zero when ¥ =o, which answers to « = 0. 

105. Forms 0)’, ~°, +1”. 

In the explicit function y= (Fwy of the variable a, 
suppose that f(x), f(x), are such, that, for the particular value | 
x =a, y assumes any one of the above forms; then, to deduce . 
a rule for the evaluation of y, we proceed thus : — 

Take the Napierian logarithms of both members of the 
equation y= (F (cs and we have 

ly = f(a) UF (2) =). 

J (2) 
Now, since, to have one of the proposed forms, #'(x),. for the 
assigned value of ~, must take one of the values 0, 00, or 1, 


lF'(x) will become either — «, + 0, or 0, and a will take 
J («) 


one or the other of the forms o and may therefore be 


INDETERMINATE. FORMS. 159 


used for calculating the true value of ly, from which we pass 
to that of the function itself. 


Ex. 1. «* forz=0 becomes 0°. In this case, BGO a 
| Ge 
which, for « =0, is equal to 
s 
4 acall = by ae Oe Geert 
1 ; 
a a T=0 
1 
Tx) 2. @2,— 0" Bee Ti 00. 
Here i) ss oe and this, when 7 =o, = bile 0: 
SiS: x 
J (2) 
site == .0 a Seal 
] 
fie 3. Cees woek ws Is 
BEE (a ke #0) "es 
ee Ranh whens 13 
fx 
bya —Il: Sea ime 


106. Form «~—~«. 
If the functions F(x), f(x), of x, both become infinite when 
a =a, then, for this value, 
E(x) —f(x#)= 0 —o. 
To deduce a rule for the Aggie) of a that take 


this form, make f(x) = F(a le \ = F(a)’ ; then the value of 


x that causes f(x), f(x), to become infinite, must reduce /}(@), 
f(x), to zero; and, if a be this value of x, we have 
pee bel Dee of @) — Fy (a), 2 2 
F(a) —f(a)= ———. : ~ 
F(a) f(a) F(a\A(a) = 
and the case is thus made to fall under the rule of a 101. 


160 DIFFERENTIAL CALCULUS. 


n 
exe; Sec. « — tan. 72 = 0 — oo when = 95, 
1 sin.a 1—sin.a 
50C.-@ —tan v7 = : — ‘ 
COsS.% CcoOs.ax COS. @ 
Sein we cos. x 
and (pe ee ice fh r =O. 
cos. a. /*=5 sm, oy 
1 2 
Hix. 2. ——-~ =0o—o whenx =I, 
lc -~ la 


dV 2\ ae a ee 
lee ny Na eS ) = 


107. It may happen that. not only do F(a), f(x), in the 
ratio aay vanish for the assigned value of the variable, but 


also all their successive differential co-efficients, however far 
the differentiation be carried. For suppose F(a) = a5 
which becomes 0 for « = 0 if a and n are positive, anda >1; 


then 
1 


nla.a 
E(x) = reas 


Wide wil. ST faa nm+1 
EY" (2) =nlaca sz (see Gee) 


Making « = I these differential co-efficients become 
z 


nlaz™ +} 


Ee (avers 


PL 


nla (nlaaro+ —(n+1) vt) 
EF" (x) a aie a a 


(8 Ie 


It is needless to carry the differentiation further to see that 
each differential co-efficient will contain a factor of the form 


m 


in which a, m, and are positive, anda >1. This factor 


an? 


takes the form 2 for z=; and if we apply to it the method 


INDETERMINATE FORMS. 161 


for finding the true value of such expressions by differentia- 
tion, differentiating p times, p being the whole number next 
above m, z will disappear from the numerator of the ratio of 
the differential co-efficients of the order p, and this ratio 


would be of the form aot in which & is a constant, and (2) 

a function of z, that becomes infinite when z=. Therefore 

all the differential co-efficients of #’(a) vanish when x = 0, 
1 

which answers to z =o. Hence,if f(x)—b «*, the terms 

E(«) | 

J(%)? 


for «= 0,if a, b, n, g, are positive, and a@ and 0 are each 


of the ratio and all their differential co-efficients, vanish 


ereater than 1. The true value of this ratio cannot then be 
found by the method of differentiation. 


ae 
When n = gq, the ratio becomes () «” the true value of 


which, for z=0, is 0, ifa>b, anda ifa< 6. 


108. The solution of cases of indetermination is often 
facilitated by transforming the example so as to make it take 
a form of indetermination different from that under which it 


presents itself. Thus 


tae | 
ae 0 
= — becomes — when xv'= 0; 
hy x W) 
but. 
1 
e # 1 1 
— =—; = ~—— when «= 0; 
a Tes 0 xX & 


and the true value of the given expression is 1 divided by. 


1 
the true value of ze when x= 0. 
21 


162 DIFFERENTIAL CALCULUS. 


Again: if (x) becomes infinite when «=o, then (Art. 


102) FC) =(F"@)) 
\ x r= ve 
But (Art. 56) 


Fe tM = FQ) ae 


and it is evident, that as x increases, and finally becomes infi- 
nite, the second member of this equation converges towards 
and finally becomes F”(x): hence 


Eoye =e Bi it es 


or by making h = 1, as we may, since hf is arbitrary, 


(= ~) = (Fw to Nie F(z)) 


If, now, the value of (F(a))=, when «=o, is required, 


we have 


(#2) =e? (1); 


a true equation, as may be seen by taking the logarithms of 
both members: therefore 


(#12) = eed (2); 


and the proposition is thus Bee to the evaluation of 


iF LF (2) ae x) 


e * , or rather nes when c=o. 


INDETERMINATE FORMS. 163 


By what is proved above, 
(=e) = Cxc seh F(z) = (: a ae (3). 


But, from Eq. 1, we have 
1 
i(sH(0) = ae ; 


therefore, by Kq. 3, 


(7): = era i 


Let this be applied to the determination of the true value of 
wo 1 
“ z= when @=oo. 
Dees 


Now, by what has just been proved, the required value is 
that of 


eee 2... fet 1\" 1\2 < 
1.2...(@+1) ira = 7 y=(+]) for c=o. 


But (Art. 9) (1 fe a et? 


109. Thus far, the discussion of the indeterminate forms 
has been confined to functions of a single variable. A few 
cases will now be considered in which these forms present 


themselves in functions of more than one variable. We re- 
mark, that a function of two variables may assume the form * 


either when a particular value is attributed to but one of the 
variables, or when both variables have particular values given 
them. An example of the first case is 

b(a —a) 
y(x* —a*) + (@— a) 


C— 


which, for <= a, reduces to "4 whatever be the value of y; 


164. DIFFERENTIAL CALCULUS. 


but by dividing out the common factor 2 — a, and then making 


2% =a, we have z= z—. 


2ay 
An example of the second is 
_ e(% —4) 
a, any DY 


which takes the form : for «=a, y =), and, for these values 


of the variables, is really indeterminate. For let p denote the 
ay G 
y— bi 
p is an arbitrary quantity, and z is therefore indeterminate. 


ratio 


then z= oe ; and, since # and y are independent, 


110. To investigate a rule for the evaluation of the inde- 
terminate forms of functions of two or more variables, take the 
function w= F(a, y), « and y being independent, and suppose 
the function to be finite and continuous for all values of x and 
y betweenz =a, x«=at+h,y=b,y=b-+k; and further, that 
all the partial differential co-efficients of the function, up to 
(n — 1)™ inclusively, vanish for «=a, y=06; but that those 
of the n™ order neither vanish nor become infinite for these 
values of x and y. 

For the time, denote by ht, kt, the increments of a and b; 
so that the function, when the values of « and y with their 
respective increments are substituted, is /’(a-+ ht, b+ kt), 
which becomes F'(a+h,b+k) by making #=1: then de- 
noting F(a -+ ht, b + kt), which is a function of ¢, by /(¢), we 
have 

Fiatht,b+kht)=f(t) (1); 
and, making in this ¢=— 0, 
F(a, b)=f(0) (2). 


If f(z) is finite and continuous for all values of ¢, from ¢ = 0 


INDETERMINATE FORMS. 165 


up to ¢= any assigned value, ¢ 7; and if, in addition, all the 
differential co-efficients of f(t), up to the (n— 1)™ inclusively, 
vanish for ¢= 0, while that of the n™ order is neither zero nor 
infinite for = 0; then (Art. 56) 


Fe Oe 8) (3) 


To simplify the application of this equation to our purposes, 


-make w’=a-+t ht, y’=b+kt; whence os h, aes =k, and 
S(t) = B(x! 9’): 
i dx dk dy’ dk 
us = 
ee ds’ ai dy! a dat” +5 ay 
Making ¢=—0, observing that then x7’—=a, y’=6), and that 
what paged! become, will be identically the same as what 
da’? dy’’ 
ee iy become when «=a, y = b, and denoting these dif 
ferential co-efficients for this value of ¢ by ) eae we 
y 
have 


P= (Ga) b+ (Gy YF 


If cat 29 , both vanish, then /” (0) = 0, and we must pro- 
da ), \dy /, | 


ceed to the 2d differential co-efficient of /(¢), which is 


OTe. " Rial 
Ody? T dyn 3 


and, in this making ¢ = 0, we have, by adopting a notation in 
harmony with that in the expression for /” (0), 


4/ VEN ak Cae Dire 
70) = (Fe) +2 (soa) tet (Gps BY 


166 DIFFERENTIAL CALCULUS. 


2 2 7 
and in this also (Gr) ... are what sans become when 
Oa hg ua? . 


x—a,y—b. If all the partial differential co-efficients of the 
second order vanish for ¢ = 0, then /”(0) =0; and we must 
pass to the 3d differential co-efficient of /(¢), and in this make 
t#=0. We should thus find 


GF iy a dF 
3 hk +3 jie it 
=) a é ay) e Gee) aay 


r= (5 


and so on; the expression for /”(¢), all up to that pene 
for ¢ = 0, being 


ft) = 

d"F ri ad a ae 

a \An Biss i a h=—1k n—1 
aa) : cp ix. (ey ) - ame err (x5 das dy" Fine hk ; 


ad" 
Rae eR 
+(+) 


the laws governing the co-efficients and exponents being ob- 
viously the same as in the Binomial Formula. 

Now, since f(x,y), F(x’, y’), differ only by having x and y 
in the one replaced respectively by x’ and y’ in the other, it 
follows that any partial differential co-efficient of F(a, y) will 
be the same function of # and y that the corresponding partial 
differential co-efficient of L(x’, y’) is of x’ and y’; and hence 
the hypothesis that renders « =a’, y=y’, will, at the same 
time, cause these differential co-efficients to be equal. There- 
fore make «=a2’=a+t+ht, y=y’=b+kt, and we may 


write 
OE 
dx a. dy 
Ret le ot ee poodle 
| 1” Tedy*— Tay : ae 


INDETERMINATE FORMS. 167 


all of the several orders of partial differential co-efficients of 
F(x, y), up to and exclusive of the n™, vanishing for «=a, 
y = b, that is, for t= 0 in /f’(t)... f-» (t); but all of those 
of the m™ order not vanishing. Then, writing #¢ for ¢ in 
Kq. 4, and substituting in Eq. 3, we have 


F(a + ht, b+ kt) — F(a, b) 


and if, in this equation, we make ¢= 1; then 
F(a+h,b-+k) — F(a, b) 


1 /d*F oF | 
123. an ae eg evar 


aCe 8 ad 
cipnog Se; dady mis eer atle +- dy a eee (5), 


y=b-+ gk 
which enunciates a theorem relating to a function of two inde- 
pendent variables analogous to that demonstrated in Art. 56 
for a function of a single variable. 
In Eq. 5, suppose both a and 6 to be zero, and then change 
h and k into y and x, as we may do, since / and & are not only 
independent of each other, but may have any values, and we 


have 


F(a,y) —£(0,0) = 


1 anh, a deh <i. 
casa lar? Sigh gon EEE k+.: . 


De Lett ater a Lie de 
-tn dady* hk + aia ae (6), 


p— phe 
which expresses another theorem relating to a function of two 
independent variables similar to that in Art. 56 for a function 


of a single variable. 


168 DIFFERENTIAL CALCULUS. 


111. Let F(x, y), f(x, y), be two functions of the inde- 
pendent variables « and y, and suppose that not only the func- 
tions, but also all of their successive partial differential co-effi- 
cients, up to those of the (n—1)™ order inclusively, vanish for 
x=a,y=b; but that, in respect to those of the n™ order, all 
do not vanish, nor do any of them become infinite for these 
values for x and y: then by Eq. 5, Art. 110, remembering 
that by hypothesis /’(a, b) = 0, f(a, b) = 0, we have 
Fiath,b+k) 


1 ad" i. ae raed 
Fila eae mit 18 age ‘dy b+ 
dni ad" ik 
-+n She sa Shee k ae 1 
dady”—} dy ea (1), 
f(a+h,b-+h) 
ay 1 d Le yn a" f n—1 
= reece ee dey 
d yee pe tl — d ad” f ken 
tw n dady"— j hk seas: dy n a (2). 
y=b-+ ok 
Dividing (1) by (2), member by member, we have 
Hiath,b+k) 


S(ath,b+k) 
d” FP, xy. kr- 1 
Ales ax De pe iy ano tang : va ; : (3). 


h* Smee n—1 iC otete EA ae L.a—1 a ay Po are 
we vt n ena: aE k- +n gig hor hk oo k | eaeeae 


Now, the increments i and k are quite arbitrary, and, like 


the variables to which they refer, are also independent of each 
other: we may therefore assume k = mh, in which m is an ar- 
bitrary constant. | 

Substituting this value for k, in Eq. 3, dividing out the 
factor h”, common to the numerator and denominator of the 


second member, and then making h = 0, we have 


INDETERMINATE FORMS. — 169 


E'(a,b) _ 
J (a,6) 
= —1 n 
~. det" ie aa ae ae ea dady"—)"" Hap a (4) 
a ai def e nt ~ : 
am a er eat Taye +9Rm jexe 
This value of Ce ae indeterminate, since m is arbi 
iS V Ta Sar 0 arbi- 


trary; and Bevferally if two functions of two independent 
variables both reduce to zero for particular values of the 
variables, the ratio of the functions for such values is really 


indeterminate. 
112. Making n=1, in Kq.4 of the last article, we have 
GB Hh 
P(a,b) _ de dy” 
a UR Re. ais 
dx my dy th 
and this value of a ; becomes determinate Tages ts ae , both 
I df 
vanish for «=a, y —); or, they remaining finite, fe dy’ ime oth 
vanish for these values of wv and y. The value of Was = : 
becomes, in the first case, 
ss dk 
(a,b) F(a,b) dx 
een As ; and, in the BORE oh) =e of 
dy dx 
oa dk 
F(a, 6) . dx = 
is also determinate if ” if we should then have 
ACAD) 7 = 
dx if 
dir df df oa ar df 
E(a,b) Ue dy" da dy 


Hana gy (Y df aa df ~ df 
dz\dx ~ dy dx dy 


22 


yi. DIFFERENTIAL CALCULUS. 


ate os 0 pe eth MN 
ibe a Bia 0, a 0) i 
Art 111, and thus have 


= 0, we make n = 2 in Kq. 4, 


which is indeterminate, except in particular cases depending 
on the absolute and relative values assumed by the partial 
differential co-efficients for the values « =a, y =D. 


at Ls le ely = 0 ay 
Example. z= oly —3~ 7 when « = Lime 
Here F(a,y) =e +ly, f(a, y) =" + 24 —8, 
TALES Wie, af 
=p" Lwor ce Sat Le ire 1, 
aN eae ‘tty a Oy oa 
Om Ty ate hence 
vee 1+m 
"Le 2m 


and therefore, for the assigned values of x and y, the function 
is really indeterminate, and may take any value between + 


and — oo. 


115, In the case of the implicit function v= F(a, y) = 0, 
we have found 


dk 

dy le 

dx NGL Le 
dy 


Now, if «=a, y = J, are values of x and y, which, while they 


satisfy the given equation, at the same time mane sly 


de 


INDETERMINATE FORMS. 171 


Bie 0, then dy takes the indeterminate form ¥ , and its true 

dy dx 0 

value, if determinate, must be found by the preceding method. 
Differentiating numerator and denominator, we have for 


r=a, y=, 
dk bed Fea dy 
dy _ _ dx uy dx? ' dady dx (2) 
dar WhO ar dy) 


dy dady dy? da | 
from which we get 


ad’? F /dy 9 Ck dy 
dy? ah dady dx 


i =0' (3), 


ae 
dy 
da 


a quadratic with respect to This equation agrees with 


Kq. 3, Art. 96, observing that by supposition 7° It must 


be remembered that Eqs. 2 and 3 are true only for the partic- 
ular values « =a,y—=b. When these values of # and y, in 
addition to making the function and its first partial differential 
co-efficients equal to 0, also make 

OE psy 02 E- ad? 


ie aia dyin. 


the value of ae as given by Eq. 2, again takes the form a 
and we must in that case effect a third differentiation, which 
gives 

oer k- dy — bk (dy ai day 
dy _ da? ab da? dy dx snare didy? Hee dady dx* (4); 
ae, er MO dy ad = (3) a2F dy 


fe 


dx* dy dady? du ' dy® \dx dy? dx? 
ad? k ar 
and from this, observing that by hypothesis ely ae = 


we derive the cubic equation, 


172 DIFFERENTIAL CALCULUS. 
aE /dy\? , dF /dy OF dy. dR 
ae 5 eee ES Dae ee ——=3 (aos 
dy* (ze) iy dady? 1¢) 1 dx*dy dx Tegal (9). 


Ex. 1. Determine the value of 2 from az? — y3 — by? = 0) 
x | 


when 2—0, y7 — 0. 


Here oye gee ene for 3,0 ae 


dx 3y?+ 2by 0 
But, for these values of « and y, we have 


yas 2ax 2a 2a = beat 
dx 3y2+ %y ty 7 ) fora = Ones 
4 5 2b — 
dx 
dy a a dan Ou i 
te Se SS ay (Se ey 
dx ,dy \du} b’ dex b 
dic 


Ex. 2. If w= a*-+ 37x? — 4a*ay —a?y?=0, find the 
value cet forx=0,y=0. We have 


sia °+ 6a?x — 4a?y,. 


dx 
a = — 4a’ax — 2a’y: 
dy 4x°+ 6a?x—4a®y  2e3 4 3924 — 2a°y 
de 4a?a + 2ary Pata + ary 


= 7 for a= 0, y =0. 


Differentiating both numerator and denominator with respect 


to « and y, we get 


d 6x? + Ba*— 20254 3a7— az dy 
Ie = dys tam a for ~=0, y=0, 
2 Pps Dy) 9 ao 
2a* + a oe ya 7 
dy 
2 eater 
ays’ 


INDETERMINATE FORMS. 173 


eg, dix dx 
dy\? d 
or a + 4 = —3=0; 
dy 
ew : 
dx vl 
Should it happen that the particular values of x and y reduce 
ak Fi SSO cal 
y to zero, while » ——, remain finite for these values, 
dy? dady dx 
d 
then Kq. 3 of this article gives, for one of the values of = 
ee 
dy es dx* 
da ere 
dady 


which is finite, while the other value of dy becomes infinite, 


da 


as may be shown by discussing the equation ax’? + bx + ¢=0, 
under the suppositions that a = 0, and that 6 and care finite. 


114. The investigation of the true value of , when it 
x 
takes the form : forz=0, y=0, may be simplified by the 


: : sages di 2 AIS Pe 
consideration, that, in this case, € ay J pcg? 28 18 eVI- 
dx pail, a y= 0 


dent from the definition of differential co-efficients. Take Ex. 
2 of the preceding article, and divide through by w?; then 


2 
xe? + 3a?— 4a? Y —a'(2) aor, 
Ms a 


. . . a ° 
Solving this equation with reference to Z and then making 
g 
e = 0, we find, as before, 


dy yy 
dx a 


Be A/T: 


174 DIFFERENTIAL CALCULUS. 


In like manner, the example 
xe! + ay*® — 2axy? — 3ax*y = 0, 
which gives 
dy 4x* — Gaxy — 2ay’? 0 
dx 3ax* + 4axy — 3ay’ Fall 
by dividing through by x’, takes the form 


2 
a (2) 2a () Sane 
2 a 2 


a cubic equation, from which, after making «= 0, we get for 


for. 2. = Uae 


Y the three values 0, 3, and —:1. 
x 


For another example, take the equation 
a+ any + bay? — y4=0; 
3 
whence a a= y43(%) — y(“)=0, 
| is 


which, for « = 0, y = 0, reduces to 
2 
he x 
and therefore we have i= 0, and Oy ae 
+ 


b 
By dividing through by v?, the assumed equation becomes 


o()+a({) +55 v=o 


which is satisfied by making simultaneously x =0, y=0, 


= 0: hence 5 zeit), orf —=o , will satisfy the given equation 


in connection with the values x = 0,4 =0. Therefore, when 


xz and y have these values, . may have the three values, 


INDETERMINATE FORMS. 


EXAMPLES. 


ogi 

( pin. x ne 
e* — 2sin. a —e7* 
( e— sin.x Ne 


eid 
gon r=? 


175 


Ans. 


a: 
cA 6 eae 
ns 6 


: 1 
In solving this example, begin by making 2? = ra whence 


1 


z= 00 when 2=—0; and we conclude that ® * decreases 


more rapidly as « decreases than does x”, however great be 


the value of n. 


5. 


1 

} n is 

i sd, a, be: ge 
n t= 0 


Pea tae 
jpn Tee). 


(f=1p(e—mF) 


a 
27 sin, — 
( * en 


‘ATSs —- 1. 


ADS. Gj Ga03...,. 


Ans. 0. 


Ans. a. 


Ans. —: 


Ans. 1. 


SECTION XI. 


DETERMINATION OF THE MAXIMA AND MINIMA VALUES OF FUNC- 
TIONS OF ONE VARIABLE. 


115. Wuen the value of a function, for particular values 
of the variables, is greater than those given by values of the 
variables immediately preceding or following such particular 
values, the function is said to be a maximum: when it is less, 
it is a muniomum. To fix attention, suppose y=/(x) to be 
such a function of w, that, as w gradually changes from a spe- 
cific value to another, y undergoes continuous changes; but, 
having increased up to a certain value, then begins to de- 
crease, or, having decreased to a certain value, then begins to 
increase. The value of y at the point where, from increasing, 
it begins to decrease, is a maximum ; and at the point where, 
from decreasing, it begins to increase, it is a munumum. . In 
the first case, the value of y is greater, and in the second case 
less, than those which immediately precede and follow. The 
terms maximum and minimum must. be understood as relative 
rather than absolute; for it is plain that a function may have 
several maxima and minima as above defined. 

Confining ourselves, for the present, to explicit functions of 
a single variable, we have seen (Art. 52) that such function 
can pass from increasing to decreasing, or the reverse, only 
when the first differential co-efficient of the function passes 
through 0 or «©, or when this differential co-efficient changes 


from positive to negative, or from negative to positive. 
176 


MAXIMA AND MINIMA. iby 


Hence those values of # which render y =/(x) a maximum 
ora minimum must be found among those which satisfy the 
equations f' (7) = 0,./’(x)= 0. | 
Let «=a be a root of one of these equations, and let h bea 
very small quantity ; then f(a) will be a maximum if /’(a—/h) 
is positive, and /’(a+/h) is negative ; but /(a) will be a mini- 
mum if /’/(a—h) is negative, and /’(a-+h) is positive. When 
fi(a —h), f/(a+h), are both of the same sign, whether posi- 
tive or negative, f(a) is neither a maximum nor a minimum. 
Eee waa 7.2); —-2an — x?, 
an) a oo ftle) =O. gives «= a; 
fi(a—h)=a—a+t+h=-+A, 
Ji(ath)=a—a—h=—h, 
Hence «=a renders the expression 2ax — x? a maximum, as 
may be easily verified; for, making «=a, 2ax — x? reduces 
to a*; but making c=a-+h, or c=a—h, our result in 
either case 1s a? — h?< a’. 
116. The method just given for deciding whether or not 
a root of the equations //(x) = 0, /’(x) =, answers to a 
maximum or minimum state of f(x), is general; but, in respect 
to the roots of the equation /’(x) = 0, we may for this pur- 
pose deduce a rule, that, in many cases, admits of easier appli- 
cation. 

As before, let «=a bea root of /’(x) =0, and suppose 
that f(s) is the first among the derivatives of /(x) that does 
not vanish for this value of x; then (Art. 56) 

fla+h)—f(a) = — f(a on) (1). 


Since 6 is a proper fraction, and h, as we shall suppose it to 


be, is a very small quantity, it is obvious that the sign of 
f™(a+ 6h) cannot change with that of h, and is therefore 
28 


178 DIFFERENTIAL CALCULUS. 


invariable: hence the sign of the second member of Eq. 1 de- 
pends on that of 4” when combined with that of f(a -+ 6h). 
But, if m is an even number, the sign of h” is positive, what- 
ever be the sign of #; and in this case the sign of the second 
member of Hq. 1, and consequently that of f(a +h) — f(a), 
will be the same as that of f”(a-+ 6h), or as that of f(a), 
since f™(a-+ 6h) and f(a) have the same sign. If, then, n 
being even, /” (a) is positive, f(a +h) — f(a) is also positive, 
whether / be positive or negative, which requires that f(a) be 
less than /(a@+h); that is, /(x),—,=/(a) is less than those 
values of f(x) which are in the immediate vicinity of this par- 
ticular value. This condition indicates a minimum state of 
the function. But, n being still an even number, if /™(q) is 
negative, then f(a+h)—f(a) is negative, which requires 
that f(a +h) be less than f(a); and a maximum state of the 
function is indicated. | 

The hypothesis in respect to A being continued, if m be an 
odd number, then, since (+-)” and (—/)” have opposite signs, 
and the sign of /”(a + 6h) does not change with that of h, the 
second member of Eq. 1 will change its sign as A changes from 
positive to negative, or the reverse. Hence /(a+h) — f(a) 
and f(a —h) —/f(a) must have opposite signs, and f(a) is 
greater than one of the expressions f(a-+h), f(a —h), and 
less than the other; that 1s, f(a) is neither greater than both 
the immediately preceding and immediately following values 
of the function, nor less than both these values; and therefore, 
in this case, =a renders the function neither a maximum 
nora minimum. Whence the rule for deciding which of the 
roots of f’(«) = 0 corresponds to maxima or to minima of f(z). 

“ Substitute the root under consideration, in the successive 
derivatives of the function, until one is found that does not 


MAXIMA AND MINIMA. ot ae 


vanish. If this derivative is of an even order, and the result 
of the substitution is positive, the root will render the function 
a minimum; but, if the result of the substitution is negative, 
the root will correspond to a maximum. If the first of the 
derivatives of the given function that does not vanish is of an 
odd order, the root corresponds to neither a maximum nor toa 
minimum state of the function.” 

Ex. 1. Find the values of x that will render 

f(x) =a? — 9x? + 24¢ —T 
a maximum or a minimum. 
f(x) = 3x? — 18x +24 = 3(x* — 6x + 8), 
fu (@)=6(@—3). 

From f(x) = 3(a”* — 6a + 8) = 0, we find « = 2, ora = 4. 
When 2 = 2, f” (x) = 6(a — 3) = — 6; and hence, for x = 2, 
the function is a maximum. Whena« =—4, /”(%)=-+ 6; and 
a2 = 4 renders the function a minimum. 

117. When y is an implicit function of w given by the 
equation F(x, y) = 0, and the values of « corresponding to 
the maxima or minima values of y are required, we may, in 
cases in which the resolution of the equation with respect to 
y is possible, employ the methods of the preceding articles. 
But, without solving the given equation, we may proceed as 


follows : — 
bet “4x, ¥) =.0; 
then (Art. 84) 


du 

AT ae dix 

dx du 

dy 
Limiting our discussion to the values of x derived from the 
equation a a) 18 fe is finite, a = 0 requires that at =f 


180 DIFFERENTIAL CALCULUS. 


Hence we have the two equations 
du _ 9, 
a 


d 


by the combination of which, eliminating y, we get a single 


Tired UE 


equation, in terms of x, which will determine those values of x 
which may or may not render y a maximum or minimum. 


2 
To decide this, we must pass to ae which, since 1 0, re- 


ay da 
duces to 
d?u 
d’y __. du? 
dac* i; du : 
dy 


and if the values of « and y derived from the equations u = 0, 


we = 0, do not cause this to vanish, but make it positive, the 
dic 


value of y corresponding to this value cf 2 isa minimum; but 


2 . 
if, by these substitutions for # and y, aig becomes negative, 


da” 
the value of y isa maximum. But, if these values of # and y 
a . ‘ oe ue 
cause to vanish, ae must also vanish in order that y 


may be a maximum or minimum; and it would be necessary to 
th ae : 

find ty and substitute in it, to enable us to decide whether 

y iS @ Maximum or a minimum. 


Ex. 1. Find the value of x that will render y a maximum or 
minimum in the function 


ux — sazy+y*?=0 (1). 


du 
dy dx ay — x? 
dz” du y? — ax 


MAXIMA AND MINIMA. 181 


du 
dx 
From (1) and (2), we find 


Pe Vag ion® = ()s 


= 0 answers toay—ax?=0 (2). 


therefore «= 0, or = ax/2, and the corresponding values 
of y arey=—0, y= an/4. 


The values x = 0, y¥ = 0, in the expression for SEN cause it 


dix 


to take the form : and the true value of a must be found. 


This may be done by the method of Art. 113. It is better, 
however, to proceed as follows: — 

The second and third derived equations of the given equa- 
tion are 


avy dy\” dy 
cae eo I a \ eee) 52: ats 
(y? — ax) za tu (ae) 2a jg t 2 = 9, 
d 
(y? — ax) 7 + (6y e- 8) 52 Y +2(5 a] 4 ig t gt 
dy 
and when, in these, we make x= 0, y=0, we find aot 0, 
d’y 2 
dx? 8a 
y=0. When aw =—aX/2,the corresponding value of y=ax/4 
is a maximum; for we have, in this case, 


Hence y = 0 is a minimum when «= 0 and 


du 
oT hea oor eid ne 2a 4/2 ae 
de du it ea K/1E — a? e/2 yas 
dy 


which indicates a maximum. 

118. Suppose that the relation between a, y, and u, is ex- 
pressed by the two equations w= F(a, y), f(x, y) =9, so 
that w is implicitly a function of a ; for, deducing the value of y 


182 DIFFERENTIAL CALCULUS. 


in terms of x from f(x, y) = 0, and substituting this value of 
y in u= F(x, y), we should have w an explicit function of «x. 
If the maxima and minima values of w were required, we 
might pursue this course; but we may accomplish our purpose 
without solving the equation f(x, y) = 0. 
We have (Art. 82) 

du eae ak’ dy 

da iG dy da’ 
also (Art. 84) 


1 du dF dFdx 
dx du dy df ; 


Now, the values of 2 and y which satisfy simultaneously the 


equations /(x, ¥) = 0, and Be 0, or the equivalent of the 


dx 
atter, 
ar df dFdf _ 
dx dy dy dx 


du 
but which do not cause dn? to vanish, will render w a maxi- 


Ge au 
mum or a minimum, according to the sign of sh ot Woks Loe, 


dix 


a2 
vanishes for these values of x and y, we must, as before 


ae; 
explained, pass on to the derivatives of the higher orders, to 


enable us to decide the question. 
Ex. 1. Given 
U=o ty? = F(a, y) (1), 
(=a) + Yb) 0 = fae 


MAXIMA AND MINIMA. 183 


to find the values of x and y that will make wa maximum or 
minimum. We find 


BME R dpi 


Te ay => Se =2(e—a), 7 = 2(y—B); 

and therefore 

atid 7 0F df: 

da dy dy dx 
becomes 

a(y —b)—y(%—a)=0: 
b 

whence ey Se, ye ao 


Substituting this value of y in oh we have 
w* — 2ax+ a? chen z+ b?—c?=0, 
2 


or Be ore ons 


a 
whence 


ee ac 
=a Vea 
By differentiation, we get from Eqs. 1 and 2 
du 2bc? 


gas fi (c? — (x — a))t 
Observing that the upper sign in the value of « answers to 


ee. au a Cage 
the upper sign in the value of — and the lower sign in the 


one to the lower sign in the other, we discover that 
ac 


CRW ay ED 


d*u ' : 
renders —— negative; hence this value of # makes w a maxt- 


da? 


mum: but, when a — 


dl? 
is substituted for a, ae. be- 


ac 
Vai +e? dae 


184 DIFFERENTIAL CALCULUS. 


comes positive, and w is therefore a minimum for this value 
OL as. 

119. When we have n variables connected by n—1 
equations, by processes of elimination, we may reduce the 
n — 1 equations to a single equation involving but two of the 
variables, and thus bring the investigation of the maxima and 
minima of the variable which is taken as dependent to the 
case treated in Art. 117. But, in general, it will be found 
easier to operate as follows: — | 

Suppose that the four variables, x, y, z, uw, are connected by 
the three equations, 

Si(@; Y; %, U) = 0, fo(@, Y, %, U) =, f's(%, Y, %, U) = 9; 
and that the maximum or minimum value of w is required, x 
being the independent variable. Differentiating with respect 
to x, we have 

df, 4 df, dy | df, dz | df, du __ 
dx. dyide i da da dita 
d df, d df, dz | df, du 
a Hf et - das F = de a 
d, fy dy c= Of, 07 ie Of, OF 
Tete ETE aot ae =?) 

One of the conditions for a maximum or minimum for u being 

Be = 0, introducing this in Kqs. 1, they become 
Of hy AY Sais eee 
das! sly das anda: | 
dj, _ afi dy df, dz __ 


dz tidy deigs de =. (2). 


df, df, dy , df, dz _ 
dat dy dzT de dz~"} 


0 | 


dy dz 


The equation which results from the elimination of —% 


dx’ da’ 


MAXIMA AND MINIMA. 185 


from Eqs. 2, together with the three given equations, which we 
will denote by f, = 0, 4, = 0, f, = 0, will determine values of 
x, y,%,and u. To decide whether any or all of these values, 
or rather systems of values, make uw a maximum or minimum, 
d?u 
dic”’ 
when the values of the variables are put init. By differen- 


we must ordinarily pass to and find what sign it takes 


tiating Eqs. 1, the resulting equations and Eqs. 1 will give a 

120. Before concluding the subject of the maxima and 
minima of implicit functions, we will briefly refer to the limi- 
tations made at the beginning of Art. 117. Resuming the 
equation 


du 

Ce ee 

SW, roma cr 
dy 


we remark, that the necessary condition for a maximum or 
1 dy on Tyr 
minimum value of y is, that 7, change its sign, which it can 
da 


do orly when it passes through the values 0 or 0. Now, 


du ne 
dy becomes zero when — = 0, -— being finite; or when 
dx dx dy 

du... ; ee et. 

‘i oe, sa being finite. Again: “2 becomes infinite when 
dy dic dix 
d d ae : du 5 
aut ee 0, o* remaining finite; or when — = ow, — being finite. 
dy x da dy 


It therefore appears that the methods heretofore given for 
determining the maxima and minima of implicit functions are 
quite incomplete, as they omit the discussion of several cases 
that may give rise to these states of value. 
Most of the functions with which we have to deal are those 
24 


186 DIFFERENTIAL CALCULUS. 


whose maxima and minima are indicated by a change in the 
sien of the first derivative when it passes through zero. It 
often happens that the conditions of the problem to be inves- 
tigated enable us to decide some of the questions relating to 
maxima and minima, which, if referred to general rules, would 


require great labor. 


EXAMPLES. 
: ie © 
rt jet eae Whenw=-,wuisa 
= — 9 
lta—2 ee 
minimum. 
x When « =1, wu isa max. 
2. et 2° F A 
+ & “« e=—Il, wis a min. 
3. u=e*-+ 2cos.e¢+e—-* Whenx=0, u=—4, a max. 
1 
4, Tie A max. when 7 =e. 


5. Divide the number a into two parts, such that the pro- 
duct of the m™ power of one part and the n™ power of the 
other part shall be a maximum. 


i) ma 
m+n mem +n 
a m+n 
their product, mn” ( ) when 
m+n 


mand nare evennumbers. The prod- 


The parts are 


Ans. 


uct may also have two mimimum 
states. 


6. Find a number such, that, when divided by its Napierian 


logarithm, the quotient shall be a minimum. 


The function to be operated with is . 


Ans, © =e. 


MAXIMA AND MINIMA. 187 


} n 
T. w=sin. x(1-+ cos. x). A max. when «= A 
x 
— , — A max. when x = cos. &. 


aay 
9. Find the number of equal parts into which a given 
number a must be divided, that the continued product of these 
parts may be a maximum. 
Pe part must be e, the number of 
Ans. 


parts “ , and the product (e)¢. 


10. Of all the triangles standing on a given base, and hav- 
ing equal perimeters, which has the greatest area? 
Denote the base by 6, and the perimeter by 2p, and one of 
the two unknown sides by «. ; 
(ein es 2p —b Ne triangle 
2 is isosceles. 
11. Of all the squares inscribed in a given square, which 
is the least ? 
Ans. That having the vertices of its angles at the middle 
of the sides of the given square. 
12. Inscribe the greatest rectangle in a given semt-ellipse. 
Let the equation of the ellipse be 
ay? b2 a? — a? b?. 
Nie fee sides 00/2, Top and its 
area is ab. 
13. Given the whole surface of a cylinder, required its 
form when its volume is a maximum. 


Represent the whole surface by 27a’. 


rani of the base Ln aXx1s ig 


Ans. : Me Ma 


volume — 


ae 


SECTION XII. 


EXPANSION OF FUNCTIONS OF TWO OR MORE INDEPENDENT YARI- 
ABLES, AND INVESTIGATION OF THE MAXIMA AND MINIMA OF 
SUCH FUNCTIONS. 


121. Let it be required to develop, and arrange according 
to the ascending powers of 4 and k, the function F(a +h, y +h), 
when F(a, y), and all its partial derivatives up to those of the 
n™ order inclusive, are finite and continuous for all values of x 
and y included between the values w anda th, y and y +k; 
h and k themselves being finite. 

For the time, replace h and k by At and kt respectively ; so 
that, when it is desired, we may pass back to h and & by mak- 


ingt=1. Then F(a+h, y +k) becomes F(x + ht, y+ kt). — 


Now, by hypothesis, x, y, h, k, and ¢, are all independent of 
each other; and, considered with reference to ¢ alone, we may 
write 

Fla+ht,y +h) =/() (1), 

F(a, y) =/f(0) (2). 

For all values of ¢ between the limits ¢ = 0 and ¢=1, it is 
evident that /(¢) and its derivatives, up to those of the n™ or- 
der inclusive, satisfy the conditions above imposed on F(a, y) 
and its derivatives. 


Hence, for such values, we have, by Maclaurin’s Theorem, 


F(t) =/(9) +(0) 5 +770) Aen cea 


ALO) pay ti) pay 


188 


EXPANSION OF FUNCTIONS. 189 


Deducing the values of /(0), f’(0), f”(0)...f(#), by the 
method pursued in Art. 110, except that now @ and y are not 
replaced by the particular values a and 8, and substituting 
them and that of f(t) in Eq. 8, we have | 


dF dk 
Fle + M, y+M) = Fley) +4 (Gh +G, 8) 


Y /@F 9 OF ae 
wae ag da a) 


i n/a kr” a ak d?f 
2 2 3 
cos (a 7 Bard: Bi, aa dady Gees ied =H) 


4- 


oR a d if dF pn-1y dF 
to (F' 195 dx" ¢ rae ike Dias ao (4). 


The ‘notations « —x-+ 6ht, y=y-+ Okt, attached to the pa- 
renthesis of the last term, signify that in the derivatives 
Gen i d°k ye 
da™ du—"dy?"* dy”? x is replaced by «+ 6ht, and y by 
y + Okt. 

In (4), make ¢ = 1, and we have 


4 


dk dk 
Fethyth=Fyt+ abt 7k 


ar 1? d? iF aa 
a 3 (qr ee Boat ay? > k*) 


1 (iF oP ar iF 
a eit DS 3 2 : a2 1 3 
+ rage +3 sage BE+ 8 Tal + Sok) 
a. 
ee, F OF, 
Bees Pee n h?- =) . 
+ 1.2.. al et mn, ao" Tas dy" ; ae T ay si er) 


y=yt Ok 
which is the development sought. 


190 DIFFERENTIAL CALCULUS. 


If, in Eq. 5, we make x= 0 and y= 0, and then in the re- 
sult write x for h, and y for k, we find 


F(a, y) = F(0,0) + Cre yi Gy )s 


i) Pare 
a*f | ar ae 
1 9 2 
T 19 ae de os (aedy) pi + (aa ‘)e 
ou y =0 pH!) 
ae . : 
OF: ae 
1 n n—1 
123 (as a" as ean Ya 
y= oy y=9Y 
af. 3 
(ae eb 
¥ =6Y 


which is the formula for the development of a function of two 
independent variables into a series arranged econ to the 
ascending powers of the variables. 

The extension of formulas (5) and (6) of this article to func- 
tions of more than two variables is easily made. For the 
expansion of M(a +h, y +k, 2+7...), we should find 
Fixth, ytkh, 2+7...)= F(a, y, 2...), 

ar, , oF 


dni h®— 1k : 
Se aah oP east (1); 


EXPANSION OF- FUNCTIONS. 191 


and if, in this, we first make a, y,z..., severally equal to zero, 
and then in the result write 2, y,2..., for h,k,7..., respec- 


tively, we have 


dF dk ak 
Flay...) = Fle yt. +(e 2 +(G (Glee 
0 
1 ar ; ar \ a2 FF: : 
eet Gert 


»(@F 
y (aay, rt 
d 


1! Een of LN, 
eg RE (=) +(a5 saat 


he grt 8 e 
oe eas Yate ts Busy, ( )3 


y=0Y 

z=62 
a formula for the development of a function of any number of 
independent variables, and in which the notation (_), signifies 
that the variables. entering the expression within the paren- 
theses are made zero. In formulas from (5) to (8) inclusive, of 
this article, the last terms are remainders expressing the dif 
ference between the value of the sum of the preceding terms 
of the development and the value of the function. When the 
form of the function under consideration, and the values at- 
tributed to the variables, are such, that, as m increases without 


192 DIFFERENTIAL CALCULUS. 


limit, the remainder decreases without limit, then, by taking n 
sufficiently great, the remainders may be neglected. 


122, Maxima and minima of functions of two or more ip- 
dependent variables. 

A function F(z, y, 2...) of several independent variables 
is @ maximum, when, being real, it is, for certain values of 
the variables, greater than J’(a+h, y +h, 2+17...); the 
increments h, k,i..., being very small, and taken with all pos- 
sible combinations of signs. On the contrary, the function is 
a minimum, when, under the same conditions, it is less than 
Pia th, ythk,2z+7...). Let us consider first the func- 
tion J"(x, y) of the two independent variables x and y, and 
endeavor to deduce from the conditions of these definitions, 
the criteria of a maximum or minimum of this function. 
tesuming Hq. 5, Art. 121, stopping in the second member at 
those terms which involve the third order of the partial deriv- 
atives of the function, and transposing f(a, y) to the first 


member, we have 


dF dF 
Fieth ytk)— F(a, y)= RECURS 


: ae ee hi avd ke? 
roa? 1 andy aye 
WY fo Ba pend d*k dr (1), 
j 3 Dee By ad 
Hore: Sane ey Rei dy mM ae 7 met 


the last term of which we fe denote by &. 


Now, if F(a, y) is a maximum, the first member of (1) is 
negative ; and therefore its second member must be negative 
also, and this whether hf and & be both positive or both nega- 
tive, or either one be positive and the other negative; and 
whatever be the values of f and k, provided only that they be 
very small. If F(a, y) is a minimum, the second member of 


MAXIMA AND MINIMA. 193 


(1) must be positive under the same conditions and limitations 
in respect to the signs and values of h and k. 

But, when f and & are taken sufficiently small, the sign 
of the pie member of (1) will be the same as that of 
Be Be tei h which must therefore be permanent and nega- 
tive if G y) is @ maximum, and permanent and positive if 
F(x,y)isaminimum. It is plain, however, that the sign of 
oi a aif will change by changing the signs of f and k. 
To make ne sign of the second member of (1) invariable, 
whether positive or oe we must have 

say me A rae AE 
and, since / and & are entirely independent of each other, this 
requires that 


dF dF 
oe = 0, and iam (2). 
Let x=a, y= 6, be values of x and y derived from these 


dad’ dF dik. 
dx*’ dxdy’ dy?’ 
respectively become when these values of « and y are sub- 
stituted in (1); then (1) becomes 


equations, and denote by 4, B, C, 2,, what 


F(a+h, b+kh)—F(a,b) = a5 (Ah? + 2B + Ok?) + R, (8). 


When the values of # and & are very small, and only such 
values are admissible, the sign of the second member of (3) 
will be the same as that of 

Ah* +- 2Bhk + Ch?, 
which may be put under the form 


Bh 
Ais (+2 at 4) 


25 


194 DIFFERENTIAL CALCULUS. 


The sign of this will be invariable, and the same as that of A, 
if the roots of the equation 


h? Bh 


are imaginary ; : being treated as the unknown quantity. 


Solving this equation, we find 
h —BtvVvB— AC, 

ke A ‘ 
from which we conclude that the conditions for imaginary 
roots are, that 4 and C have the same sign, and that the prod- 
uct AC be greater than B’. 

In recapitulation, we say, that if «= a, ¥ = 6b, make F(a, y) 
a maximum then for these values of x and y we must have 


dk dF 
dx he dy =a 
ea ae both negative 
Magee Aah 8 , 
Cra a? k\? 

ee dx? dy? Ga) ; 


If c=a, y —), make F(x, y) a minimum, the conditions are 
2 


a hed MK oh 
The existence of real roots for Eq. 4 indicates that we may 


the same, except that then must both be positive. 


give such signs and values to f and & as to cause the expres- 
sion Ah? + 2Bhk 4- Ck* to vanish, and, in so doing, change its 
sign, which is incompatible with the existence of a maximum 
or a minimum state of f(a, y). 

123. There remains to be examined the case in which 
AC— B*?=0. When this condition presents itself, there may 
also be a maximum or minimum value of the function. 


MAXIMA AND MINIMA. 195 


By the theory of the composition of equations, or by in- 
spection, the expression Ah? + 2Bhk + Ck*® may be written 


Ie? h 2 alu 
5 (45 +2) Je oN t 


aren ye 
hence, when AC — B? = 0, this becomes q(45 +B) , the 


h ; é 
sign of which, except when (4 7" —- B) vanishes, is the same 
as that of 4; and F(a, b) is a maximum if 4 is a negative, 


eta ~ h : 
and a minimtim if 4 is positive. Should (4; + 2) vanish, 
as it does when ae pps cannot tell, without further 
inquiry, that M(a+h,b+k) — F(a, b) does not undergo a 
change of sign. To decide this, let Z, M, N, P, represent the 
I P ove Hy Baek: <a i ely ok ae te 
values of ——, datdy? dady®, dy®’ respectively, when «=a, 
y —b; and also put #2, for the value of 


4 + 4 
(ah ‘4 od qh $4 ie 5) 
ae 


y=y+ 9x 
for the same values of x and y. 


Introducing these values, and the conditions 


ar _,, a 
ody 


= 0, Ah? + 2Bhk + Ck=0, 

wee. h B 
the latter being a consequence of the hypotheses t=-Z 
and AC’ — B? — 0, we have 
Fia+h,b+k) — F(a, b) = 


rag (Lit + 8M + SNAG + Ph) + By, 


196 DIFFERENTIAL CALCULUS. 


From this we see, that since the sign of F(a +h, b+k) — 
F(a, b), when h and & are very small, is the same as that of 


Th? + 3M h?k + 3Nhk*? + Pk’, 
F(a, 6) cannot be a maximum or minimum, unless, if Ay +B 
vanishes, 
Lh? +- 3M h?k + 8Nhk? + Pk? 
vanishes simultaneously. Suppose these conditions to be sat- 
isfied, then the sign of F'(a-+h,b-+k) — F(a, 6) is the same 
as that of &,. But we have shown, that when 4 + B was 


not equal to zero, and # and & were taken suteienae small, 
the sign of F(a +h, b+ k)— F(a, 6) is the same as that of 4; 
but, when /’(a, b) is a maximum or minimum, the sign of 
F(a+h,b+k)— F(a, 6) must be invariable for all values 
of h and & which are small enough to cause F(a, y) to change 
in value by continuous degrees in the immediate vicinity of 
the value F(a, b). Hence it follows, that, when the values 


of h and & are such as to make : oe —3, these values must 


give #, a sign the same as that of A. If these several condi- 
tions are satisfied, the function is amaximum when 4 is nega- 
tive, and a minimum when 4 is positive. 
124. If d=0, B=0, C=0, then 
F(a+h,b+k) — F(a, b)= 
1 

1.2.3 
L, M, N, P, R,, denoting the same values as in the preceding 
article. Hence, in order that F(a, b) may be a maximum or 
minimum, L, M, N, P, must separately vanish, and the sign of 
f#, must be invariable; and generally, when F(a, b) is a maxi- 
mum or minimum, all the partial derivatives up to those of an 


(Lh? + 3Mh?kK+3Nhk? + Pk) + R,; 


MAXIMA AND MINIMA. 197 


odd order inclusive must vanish; while those of the following 
even order are so related, that the sign of the expression in 
which they are the co-efficients of the powers and products of 
h and k remains the same, whatever be the signs of / and k. 
The conditions which will insure this invariability of sign 
when the derivatives are of a higher order than the second, 
are, in the general case, too complicated to be here discussed, 
or even to be of much practical value. 

125. Let it now be required to find the maxima and mini- 
ma values of /’(a, y, z), a function of the three independent 
variables x, y, 2. 

Referring to Eq. 7, Art. 121, and in the second member 
stopping with the terms involving the partial derivatives of 
the third order, denoting the aggregate of the remaining 
terms by f#, and carrying F(x, y, z) to the first member, we 


have 
Fia«th,y+kh,2z+7)— F(x,y,2) = nh + Skt ei 
1 /@F Se is 
+ra(ga" toa eet daly os ake 


do Liye 
+ 2-—— yas ki) +f (1). 
When h, k, 7, have the very small values which alone are 
admissible in our investigation, the sign of the second mem- 

ber of (1) will a, ag that of ae expression 

oh a dF ? she! 
dy 

and the sign of e.. thy tk, z : 1) — F(x, y, 2) cannot 
remain invariable for all the possible values and combinations 


of the signs of h, k, 7, unless 


dk’ 
dz” + “pet i= 0; 


198 DIFFERENTIAL CALCULUS. 


which is, therefore, the first condition necessary to insure a 
maximum or minimum of /'(x, y,z). But, because h, k, 7, are 
entirely independent of each other, the above condition in- 


volves the three following : — 


aE dk dk 

ae ») 

dx ’ dy = 0, ve ie = 0 (2), 
which will determine one or more systems of values for a, y, z. 
And we now proceed to inquire what further conditions must 
be satisfied when one of these systems, say x =a, y=b, 2=c, 


renders the function a maximum or minimum. 


Substituting these values in (1), and representing the val- 
BE fad ets MN MCs et Meas ae RP 6 RM Da i 
ues which ——-, =, Ft, then assume, 


de®’. dy?’ d2*? dady’ dadz’ dydz 
by A, B, C, A’, B’, C’, B,, respectively, we have 


Fia+th,b+k,e+1) — F(a,b,c) = 


= (Ah? + Bk? + Ci? + 2A/hk + 2B/hi + 20°) +R, (8), 


the sign of the second member of which, when h, k, 1, have 


very small values, is the same as that of the expression 
Ah? + Bk? + Ci? + QA‘hk + 2B/hi + 2C'la (a); 


and this sign must be permanent during all the changes in 
the signs and values of h, k, 7, if #’(a, 6, c) is a maximum or 
minimum of /'(a, y, 2). 


Expression (a@) may be written 
(45 es t+ 242 oe 7 +207). 
Make s= is hom, a then we have 


i?(As? + Bt? + O + 2A’st + 2B’s + 20't); 


MAXIMA AND MINIMA. 199 


and the sign of this last will be the same as that of 


As? + Be . C+ See + 2C"t, 
or A(# = aE Fat 27 bry C 7‘) (b). 


The conditions that will make the sign of (b) invariable for 
all possible values of s and ¢ will make that of (a) invariable 
for all the combinations of signs, and all admissible values of 
eeents 10 find these conditions, put (b) equal to zero, and 
solve the resulting equation with UE ge) to either s or 7, 


say s. We thus get 


2 eee avi (A”? AB) P42(A'BI—AC)t+B *—ACh. 


vi. if the quantities A, B, C, A’, B’, C’, have such relative 
values, that the quantity under the radical, in this value of s, 
cannot become positive for any real value of ¢, then the paren- 
thetical factor of (>) will always be positive, and the sign of 
(b) will be the same as that of 4. Putting this quantity un- 
der the form 

At B— AQ’. BU AC 
(Av AR) G Rema, Fes Fy) 
we see, that, to make it negative for all values of ¢, it is neces- 
sary and sufficient to have 
A” — AB< 0, ie., AB—A”>0 (c). 
@4 8 = AC"? < (BY — AC) (A"* AB) (d). 

When conditions (¢) and (d) are satisfied by the values of 
x,y, #, deduced from Kqs. 2, (a, b,c) is a maximum or a mini- 
mum; for then the sign of expression (0), and consequently 
that of the second member of Eq. 3, is permanent, and the 
same as that of d. (a, b,c) is therefore a maximum, if .4 is 


negative; and a minimum, if A is positive. 


200 DIFFERENTIAL CALCULUS. 


Hence the conditions necessary for the existence of a maxi- 
mum or minimum of F(a, y, z) are, that the values of a, y, z, 
derived from the equations 


dP, ak _ 9 CF _ 0, 
da ie ij rre, eae 


should make 
O73 2k dad? 
dx? dy? Se dy 


) > 0, that is, positive (c’); 


and 


Pd Fore ieee 
dxdy dxdz da? ae 


ar a CF) (OCF Cr ir di) 
(Saas) ~ da* dz* ) (say) ~ dx? dy? er 


A necessary consequence of conditions (c’) and (d’) is, that 


Bd LO ae es oF GE |) fai SO: 
(ae dx? ie oe * da? da? —( Fa 
Cr PF ar 


and hence da? dy? di must all have the same sign, which 


is negative when F(a, y,%) is a maximum, and positive when 
F(a, y, 2) 1s a minimum. 

176. If we have a function F(a, y, 2...) of m independent 
variables, the first condition for the existence of a maximum 
or minimum would be 


Git ak es eee |, 


whence, because h, k,i..., are pia of each other, 
dk A Bh on he 
a 0 ra age = e 
dx ’ dy ’ dz wee 


Kqs. 1 determine values x = a, y =b, 2 =c..., which may 
or may not produce a maximum or minimum state of the func- 


MAXIMA AND MINIMA. 201 


tion. To decide this question, we should have to examine the 
term 


1/@F_, @F., @PF dF 
| Sy ee a aay SIE i dare seks 
mites ne era Tapa, eG ) 


in the expression for 


F(athytk, z+...) —F(a, y,2...). 

If, for these values of x, y, z..., the sign of this term is per- 
manent, and negative for all admissible values, and all the com- 
binations of the signs of h, k,7..., the function is a maximum: 
if the sign is permanent and positive, the function is a mini- 
mum. It would be found, that, to insure either of these states 

meee. Oh dei 
of the function, aie dak 


sign, negative for a maximum, positive fora minimum. But 


-++, must all have the same 


the investigation of all the conditions to be satisfied in this 
general case, in order that the function may be a maximum or 
minimum, is too complicated to find a place in an elementary 
work. 

127. Maxima and minima of a function of several variables 
some of which are dependent on the others. 

Let it be required to find the maxima and minima values of 
the function u= F(x, y, 2...) of the m variables a, y, 2..., 
which are connected by the n equations | 


i EQUAL ee at 
Bete Wen 


Fl; Y; 2.) —0 | 
By eliminating from uw, n of the m variables, by means of the n 
given equations, «would become an explicit function of m—n 


independent variables, and its maximum or minimum could 
26 


202 DIFFERENTIAL CALCULUS. 


then be found by the method just explained; but this elimi- 
nation may be avoided, and the determination of the maxima 
and minima of w greatly simplified, by the process which fol- 
lows : — 

Suppose the variables 2, y, z..., to receive the respective 
increments h, k,7..., by virtue oft which the function passes 
from a given state of value to another in the immediate vicinity 
of this: then, if the given state be either a maximum or a 


minimum, we must have 
LE dF , Ba): 
yf a ee dz v fe MEAN or ( ). 


The partial derivatives of the first members of each of 
Kas. 1, taken with respect to each of the variables, are separate- 
ly equal to zero. Taking these equations in succession, multi- 
plying each partial derivative by the increment of that varia- 
ble to which the derivative relates, and placing the sum of the 
results equal to zero, we have 


apie Aare ee 
Ving Hep His 20 
ny if, dfy . *. 
ip a i a 
te ME es? 

iy Ae ian 4 


There an a ea (nx) of Eqs. 3, these with (2) make n + 1 
equations of the first degree in respect to the m quantities 
h,k,v...: hence, by the combination of these equations, we 
may eliminate n of these quantities, and arrive at a final equa- 
tion involving the remaining m — n quantities, and also of the 
first degree with respect to them. To facilitate this elimina- 
tion, let My, Hy...,4,, denote undetermined quantities; and 


MAXIMA AND MINIMA. 203 


multiply the first of Eqs. 3 by »,, the second by p,..., the 
nm by u,; add the results to (2), and arrange with reference 
toh, k,1...: we thus get 


df dfs Ta 7, 
ate By at a a rel tela ead a a 
df, df, df, 
+(a, +8 oa gp tir ol aols ze | 
dF df, df, 
eG fy tb Ho Foe a 2 
+ ° . . e 


a true equation when F(x, y, 2...) is susceptible of a maxi- 


0 (4); 


mum or minimum, whatever be the values of p,, M2. ..-, Ma: 


Place the co-efficients of n of the quantities h, k, 7..., in 
Hg. 4, equal to zero: the m equations thus obtained will deter- 
mine ft), /)...,“,. By substituting these values of (1), fg.-+) Mn; 
in (4), 2 of the quantities h,k,7..., will vanish from that equa- 
tion; and, if the co-efficients of the m—n of these quantities 
remaining in the equation be placed equal to zero, we have, 
including the n given equations, 


Re a, ital irae te eh oy Uf 8s) 0, 

m equations from which to determine values a, b, c..., for the 
m quantities x, y,z..., respectively. This is equivalent to 
equating the co-efficient of each of the m quantities h, k,7..., 
in (4), to zero; and these m equations, together with the n 
given equations, will make m+n equations, by means of 
which we may eliminate the n indeterminates py, ,..., M,, 
and find the m quantities x, y, z 

It remains to be ascertained whether the sign of the expres- 
sion for (a+h,b+kh,c+7...)—F(a,b,c...) is invariable; 


and, if so, whether it be positive, which answers to a minimum; 


204 DIFFERENTIAL. CALCULUS. 


or negative, which answers to a maximum. Theoretically, 
this examination is very complicated; but, for most cases in 
which this method is applicable, the form of the function en- 
ables us to decide at once.which of the two states, if either, 
the function admits. 

When m — n=1, or there is only one more variable than 
there are equations connecting them, the case discussed in this 
article reduces to that of an expression which is implicitly a 


function of a single variable. 


128. In the case in which it is required to determine the 
maxima or minima of a function, the several variables of which 
are connected by but one equation, the process may be still 
further simplified. 

Let u= F(x, y,2...) be the function, and 

J(&, Y, %---)=0 (1), 
the equation expressing the relation between the variables 


x, y,%...: then, by the reasoning employed in the last article, 
we have 


eht+ 5, bp Se 9 ay 


e 
hae t, ke t, i+. (3). 
Multiplying a by Re eee vu, subtracting the result 
from (2), and arranging with reference to h, k,7..., we have 
a | hea aE a: dB) ape Ae 


Kquating to zero the co-efficients of the several quantities 
h, k, v..., we should have, with the given equation, m +1 equa- 
tions, by means of which we can eliminate », and determine the 
m quantities. But from the co-efficients of h,k,7..., in (4), 
placed equal to zero, we find 


EXAMPLES OF EXPANSION. 205 


dF dF dF 
eS _ dy: anes 
Ga a aie 
da: dy dz 


that is, the ratio of the co-efficients of h, in Eqs. 2 and 3, is 
the same as that of the co-efficients of k, of 7... . This rela- 
tion will be found to facilitate the determination of maxima 
and minima. : 

The examples which follow are arranged in the order of the 
articles in this section under which they fall. 


EXAMPLES. 


1. If F(a, y) = «?(a+y)’, find the expansion of 
| (w@thy(aty +k} 
in the ascending powers of / and k. 
(2?(a-+-y)'+20(a-+y)h-+ 30%(a-ty)%h 
+(aty)h-+6e(a-+y)%hle-+3a%(a-+y)e? 
+3(a+y)?h?k + 6a(a+ y)hk? + x?k? 
+ 3(a + y)h?h? + 2xhk® + h?k?. 
2. If F(a, y, 2) = ax® + by? + cz? + Lexy + 2guz + 2 fy2, 
find the expansion of F(a+h,y+hk,z-+7). 
an? + by? + ca* + Lexy + 2gxz + 2fyz 
+ 2(aa + gz + ey )h+ 2(by + fa + ex)ke 
+ 2(cz+fy + ga)i+ (ah? + 0k? + ci?) 
ee 2(fki + ghi + ehk). 


(wh) (aty+k)= 


Pathy+h2to= 


3. Expand 
2 
(atarg tari: )(bo tot +? AEE -) 


in the ascending powers of x and y. 


206 DIFFERENTIAL CALCULUS. 


a ee 2 
(a+ a, i tars + \(b4 64 +b + +) 
=A by + box + aydyy 
1 
de 1.2 (A,b)x? + 2a,b, xy + a,b,y”) 


‘| 
+ 1.2.3 (a,b) x° + 3a,b,x°y + 3a,b, xy? + aobzy’) 


foe 
4, Find what values, if any, of w and y, will render the func- 
tion F(x, y) = wy + xy* — axy a maximum or minimum. 


From the equations ae carl WY a = 0, we get four systems 


dy 


of values, viz. 


31) mayb emer 1 i | a | 
3 
) ? ) } 
y=0 Hot a yas 
3 
none of the first three of which satisfy the condition 


ee 


2 
oC ee ae males 
oi tyare ) (Art. 123), 


dady 
and must therefore be rejected. The fourth system reduces 


Be se F cok Poe ae tite q he 
is inequality to 9° ny which is true, and at the same 


74 
2 2 : 
time makes both Es and ee positive: hence the values 
on dy? 
z=%, y =o make the function a minimum, and this mini- 
a’ 
mum 1s — 57° 


5. Determine the values of « and y that will make 
Eo y) — en F(a Boa 


a maximum or minimum. 


EXAMPLES.— MAXIMA AND MINIMA. 207 


bam |) dF 
dy 


dx 
oo -{) fl emed ea—-t1 
at ae} a0 b 
which we will examine in succession. 


Pr ar a 
The first system gives Tx? Say Hiatal dears 


hence the values.« = 0, y = 0, make the function a minimum. 


= 0, give the three systems of values 


With the second system, we have 


al eT SI fee Ls ae 

dx? = 2(a — be", as — — 4be7!, ee 
hence the existence of a maximum or minimum depends on the 
; i at 
relative values of a and 6b. Ifbis greater than a, —,, =~’ 

ain aia 
have the same sign, which is negative, and the function is a max- 
2 2 

imum; but, if b be less thana, ee at have opposite signs, 


and the second system of values of x and y make the function 
neither a maximum nor a minimum. 
For the third system, we find 
2 2 2 
= — 4ae, at = 2(b — a)e™", ia Sa fe 
from which we conclude that = +1, y =0, will make the 
' function a maximum when a>b; but, when a< J, it has 
neither a maximum nor a minimum. 
6. The equations of two planes, referred to rectangular co- 
ordinate axes, are 
Ji(%, Y, %) = Av + By + C2 — D=0 (1), 
PA, Y, 2) = Ale + Bly +C'e—D'=0 (2). 
It is required to find the shortest distance from the origin of 
co-ordinates to the line of intersection of the planes. 
Let F(a,y, 2) =a? fy? +22 (3) 


represent the square of the distance from the origin to the 


2.08 DIFFERENTIAL CALCULUS. 


point of which 2, y, z, are the co-ordinates: then, if a, y, 2, 
are the same in the three Eqs. 1, 2, 3, the concrete question is 
reduced to the abstract one of finding the values of a, y, 2; 
which, when subject to the conditions of Eqs. 2 and 3, will 
render f(x, y, %) @ minimum. | 
By Art. 128, we have 

20 + p,A + p,A’=0 

2y +4,B+¢,B/=0 ¢ (4). 

22 + yO + pC’ = 0 
Multiplying the first of Eqs. 4 by A, the second by B, and the 
third by C, adding the results, then, by (1), we have 

(A?-+B? 4 Ou, + (AA + BB+ OO), +2D=0 (5). 
In like manner, 
(A + BV+ CO”), + (AA + BB’ 4+ CO) pu, 4+ 2D'=0 (6). 
From Eqs. 5 and 6, we get the values of »,, u,; and Eqs. 4, 
when these values of ,, u., are substituted in them, will de- 
termine x, y,z. Multiplying the first of (4) by a, the second 
by y, and the third by z, adding results, and reducing by Eqs. 
1, 2, and 3, we have 
2F (a, y, 2) + Dey +D/p,= 0; 
from which we get F(x, y, 2). In this case, it is unnecessary . 
to examine the sign of F(x-+h, y+hk, c+1) — F(a, y; 2), 
when the values of a, y, z, are substituted; for we know from 
the conditions of the geometrical question that the function 
has a minimum. 
7. Required the values of a, y, 2; that will render the func- 
tion 
Ue — aes Fee 

a maximum, the variables being subject to the condition 


yv=ax+byt+ecz—k=—0. 


EXAMPLES.—MAXIMA AND MINIMA. 209 


We find 

Ci BAF os | fy isd) Ute wae 

cas! taal y ra U, dy y dz” 2 ’ 

dv dv 7 i ; 

ome dy, de 

therefore (Art. 128) 
VA) 8 GINS al 
aby Cz I ; 
p k q k r k 


Se - nae Oe ce aera LA oo patacreman per prea 
peer 9 bp +g+r Ra rN! fn ef 
These values of x, y, z, make the function a maximum. For 


we find 


en de ct” dy? y dy ye dz? 


du __ Clase 


Goad. . Oe GL os au  redu 
ee : i a dd | a8 


and these, because ‘= 0, for the values of 


dx ’ dy dz 
x, ¥, %, become 
ere i AO Qe int r 
0 de tm aa 
all of which are negative, —a necessary condition for a maxi- 
du du dey 


mum; and, by getting the partial derivatives diedly dadz’ dydz 


we see that the other conditions (Art. 126) to insure this state 
of the function are also satisfied. 

By making a= 1, b=1, c=1, the above becomes the solu- 
tion of the problem for dividing the number k into three such 
parts, that the product of the p power of the first, the g power 
of the second, and the r power of the third, shall be a maxi- 
mum. 

8. Inscribe in a sphere the greatest parallelopipedon. 

If a be the radius of the a the parallelo- 
Ans. ; 


pipedon is a cube having —; ai ~ for its edge. 


27 


210 DIFFERENTIAL CALCULUS. 


9. Determine a point within a triangle, from which, if lines 
be drawn to the vertices of the angles, the sum of their 
squares shall be a maximum. 

The point is the intersection of the lines 
Ans. 4 drawn from the vertices of the angles to 
l the centres of the opposite sides. 

A function F(a, y,...) of two or more variables may be of 

such form, that it admits of a maximum or minimum for values 


dF df 


of the variables which make —_, ——,... indeterminate or in- 
dx’ dy 


finite. There are no general rules applicable to such cases; 
but each one must be specially examined. 
10. What values of x and y will make 
wu = ax? + (x? + dby?)8 
a minimum ? 
LE Sy yall 2 : du _ 2 y 
dx 3 (a? + by*)8 1) (a? + by?)8 
For «= 0, y = 0, these differential co-efficients take the form 


3 but their true values are infinity ; for, if we make y = mz, 


they become 
eae 


dx 


1 du 2 m 
af (1 + m2b)® dy 3 a8 (1 + m2b)3- 
Hence for « —0, and therefore for y= 0, at the same time, 


=2an +5 


we have 

du 

Te O, dy =—=,00); 
For «= 0, y = 0, we have uw = 0; and no real values of # and 
y can make w negative. Hence w is a minimum for «=0, 


Uieoe Us 


SECTION XII. 


CHANGE OF INDEPENDENT VARIABLES IN DIFFERENTIATION. 


129. It is often required in investigations to change dif- 
ferential expressions, obtained under the supposition that cer- 
tain variables were independent, into their equivalents when 
such variables are themselves functions of others. 

Suppose that, having given y=/(z), c= q/(z), it is required 
to express the successive derivatives of y, taken as a function 
x, in terms of those of w and y taken with respect to z. 

We have found (Art. 42) 

dy dy dz 
Gai dz ds’ 


and (Art. 41) 


dy 
Cau lor ty. dy dz 
JOD EER GG We tase 
dz dz 
dy dy 
gy ad’ da d dz dz 
eecS dat = da de — da du dz ‘AT *) 
dz dz 
d*ydx dx dy 
dz? dz daz? dz dz 
= a ae . ao 
d 
iy dx da dy 
_@da dedi da 1 
0 5a de dee: 
(e) i 


211 


yl DIFFERENTIAL CALCULUS. 


So also 


dy dx  d’x dy 
ey) dda? de og) oe 


dx? dx da\* 
=) 


d’ydx  d’a dy 
_ addz? dz dz? dz dz 


rae. Vda ®t TS de 
(i) 
Ge dx da a) (=)- a(T) 7 (at da d? x a 
dz? dz dz* dz/\dz dz / dz* \dz2 dz  dz* dz/dz 
7. ap. uy. fda\®, 0b): cn 
a, : 


(a axes O°2 ) a wa (ae de .d*m Z) 


dz? de dz> dz/dz “dz? \dz* de daz? daz 


-, dax\* ‘ 
dz 


4, 5 
In the same manner, we may find es , ee .. Substitut- 
nay Lata 
dz’ dz’ dz? 


ing in these the values of -++, found from 


y =f(2%), c= 9(2), 


we have the values of the successive derivatives of y 
with respect to a, in terms of those of « and y with respect 
to %. 


130. Having y =/(x), to change the independent variable 
dy d*y 


from x to y in the expressions for — 


daa! jn 


: ee ees eee 


CHANGE OF INDEPENDENT VARIABLE. 213 


1 
Since aa da (Art. 41), 


Meh dydads vir 


dy dy 

d*x da 

mo ey dy et Jay? 

Ae nee 

dy a) 

Similarly, 

d*a ete 

dey Hi dy a a ae 


raf 3 (= a\? 
dy? 7 ae (es ) 
hee: ul 


ie dx 
iy) 
d®a da d?a\? 
— — dy* dy a) 


1 ‘ ay a4 
In like manner, we may find the expressions for dpe? a3 a 


These formulas may also be found from those in the preceding 
article, by making z = y; whence 


dy _ ae a 6 diy _ 4 ea Ce Ocoee 


Dieter’ 


C7 mer a2" ATS: EPR ae ci dake dy 
By the introduction of these values in the formulas of Art. 

129, they will be found to agree with those just established. 
131. Having given iar Oy GL), 


and also eecus. 0, 77 sin.§ (2), 


214 DIFFERENTIAL CALCULUS. 


it is evident that we may eliminate x and y from these equa- 
tions, and get a direct relation between r and 6; and thus r 


becomes a function of 6. 


dy a? : 
It is required to express the values of ae a" -» derived 
Ce ae: 
from Kq.-1, in terms of —- SPU 


By Arts. 41, 42, we have 
dy 
dy dydo dy1 _ do 
da di dx dé'da™ (da 


do dé 
sin. 6 ia + 7 cos. 6 
= 8 from Eqs. 2 
cos.6 ie 7” sin. 0 
also ; 
sin. 05 abr COs sin poo 
dy da a TES ole "dé 
dx? da +e on dx 
cos. 0. — resin. 0 . cos. 7, — r sin. 0 


Performing the indicated differentiation, we find for the nu- 


merator of the result 


RTA IP Meade ie ot eee 
sin. AT acm COs. Pee: sin. Ji eos 35 ee ) 


d*y Py woe dr 
—( cos. 6 aan eater me PE sin. 0 Tse : 


dr d?r 
hich cme § 
which reduces to p29 (a) - 7 Jp?! 


‘ . ae dr 
and the denominator, remembering that — —cos.6é— —rsin.6, 


dé do 


ae ee ee 
COS. ee fe ASB . 
do 


is 


CHANGE OF INDEPENDENT VARIABLE. y Mies 
; dr\? d?*r 
eds 42(2) ml Se Ey" 


da? ew, Ae dr ; : 3 . 
a) 


These formulas are used in the applications of the differen- 


Hence 


tial calculus to geometry, where a change of reference is 


made from rectilinear to polar co-ordinates. 
132. Suppose that we have the expressions for aie ae 
dic - dy 
found from the equation vu = F(x, y); but that the variables 
«and y are connected with two other variables, 7 and 6, by the 
equations x = f(r, 6), y= F,(r, 6): then we may conceive 
x and y to be eliminated from these three equations, and w 
to be a function of r and 6. Required the equivalents of 


the expressions for na oe in terms of the derivatives of 
das ody | 
x, y, and w, with respect to r and 6. 
By Art. 82, we have 
du __dudx , du dy ) 
dr ~ dx dr ' dy dr | 


(1); 
du _dudx _ du dy 
idee a dy do. 
and from these two equations the values of an pes can be 
| Cone a1) 


found. 

When the equations expressing the relations between 
Ueeemoware 7 — I(x, 7), 0 = Fi (x, y);, mstead of those 
given above, then 

du _dudr , du da ) 
dz dr dz + di dx | 
du _ du dr, dud | 
dy dr dy ' dé dy | 


(2). 


216 DIFFERENTIAL CALCULUS. 


If the variables x, y,7, 6, are connected by the unresolved 
equations F\(z, y, r,6)=0, F(x, y, r, 6) = 90, we proceed 
thus : — 


By Art. 82, 
al cca Oe diy dy _ 
db} de (de dy dia 
dF, dl, ds diy dy am 
Gokia da dy ao} 
in which it must be remembered that (a) (=) are par- 


tial derivatives of /,, #,, with respect to 6. 


Differentiating /,, F,, with respect to 7, we get two similar 
equations involving oe ; and the four equations thus ob- 
. » 04 dy “dawrdy mene | 
tained will determine a?) Hat ieee hich must be sub- 
stituted in formulas (1) or (2), Art. 131. 


The following example will illustrate the manner of using 


the above formulas : — 
Given u=/(x, y), «= 7 cos. 0, y= 7 sin: 0, 1b Ieee 
du du du -— du 


to express in terms of 7, 0, 


a Mey’ dr? das 
We have 
dx dy 
Ap 6, q9 == 7 COBO; 
dx dy 
a i] aw POs 
cp cos.4, ae sin. 0 
Hence, by formulas (1), 
du ; 
A ee ee + sin. 6 = 


CHANGE OF INDEPENDENT VARIABLE. 217 


du Cine roe du | 

whence Tp = 008 OG — Bin. 05, | 
r (a). 

rey jes Ue ee | 

ere digeany d0) 


To make formulas (2) applicable to this example, we first 
deduce, from the equations x = r cos. 6, y =7 sin. 6, the values 
ef r and @ in terms of wand y. We find 


— Jar + y?, 6.== tan.>} e 


whence dr_«a dr_y ddO_ _y @_&, 
dx” r’ dy. r’ der?’ dy r?- 


and, by means of these, Eqs. 2 become 


du _ «du y du 
dze™ r dr’. r* do 
du _ydu, « du 
dy rdr' r® do 


. . e Hb; 
aoerelations. « — 7 cos. 0, y = 7-sin. 0, give cos. 0=-, 
A 


(b). 


sin. 0=% , by means of which we can pass from formulas (0) 
to (a), or the opposite. 
133. Attention is here called to the necessity of attaching 
their precise signification to the symbols 
pemarivds dy i dd} dio du dy 
dz’ dy’ dr’ dr’ dx’ dy’ do’ do’ 
which occur in formulas (1) and (2), Art. 131. 
It must be borne in mind that these denote partial differ- 


ential co-efficients, and that those referring to the same varia- 
ai da 


bles, such as —_, , have not to each other the relation of 
dx dr 
to a which are derived from the equation f(x, y) = 0. 


With reference to these last, we know that one is the recipro- 
28 


ca 


218 DIFFERENTIAL CALCULUS. 


cal of the other, or that their product is 1; but this is not 


true for ue x ae . The consideration of the meaning of the 


dr 
term “ differential co-efficient,” and the difference between the 
equations connecting the variables in the two cases, will re- 
move all difficulty. In getting formulas (1), x and y were given 
as explicit functions of the independent variables r and 6; and 


a change in either r or @ will produce changes in both # and y. 
Hence, in the operation of finding oe r, c, and y vary, while 


0 remains constant. In formulas (2), 7 and @ were given as 
explicit functions of « and y; and a change in the value of 


either « or y will produce changes in both v and 9; and hence 


Port! 
the increment attributed to « in getting “” causes 7 and 0 


dx 


also to vary, while y remains constant. ‘That is, in formulas 


| 


fan, zs supposes 7, x, and y to vary together, 0 being constant; 


while, in formulas (2), - supposes a, 7, and @ to vary, while y 
remains constant. Thus it appears. that these two partial de- 
rivatives are obtained on different suppositions in respect to 
the variables which receive increments, and those which 
remain constant. 

In the example just given for formulas (1), we have 


= = cos.0; and, for formulas (2), i. = cos. 6; and the product 
Th a inate 
fae 

154, Having u= F(a, y, 2), and three equations express- 


ing the relations between a, y, z, and three other variables 
du du, du-:: 

dz’ dis da?” 
terms of the different co-efiicients of w with respect to 7, 0, w. 


vr, 0, w, it is required to find the values of 


CHANGE OF INDEPENDENT VARIABLE. 219 


By Art. 82, 
du __duwda , du dy du dr 
dx dd dx 2 dw dec ' dr dx 
du dudod .dudw . dudr 
Aiiee 1): 
Se deay » dw dy © dr dy (1) 
du _dudd, dudw , du dr 
da @ dat dy dat dr a 
The three equations connecting a, y, z, 7, 0, w, will enable 
ag do) - da dr dw 
ee. dy’ Hae yen) Wee and Eqs. 1, when 


these values are substituted in them, give us the expressions 


us to determine 


sought. 
du du du 


da’ dr’ dw’ 
du du du 
expressed in terms of eas zi OF We may find these 


By solving Eqs. 1, we can also find the values of 


values from the equations 


Gia duds Jjduidy > du. dz 
do dx do dy do ' dz do 
du. dudx dudy du dz 
ee ieee Midi si deductions”: 
gp du de du dy du dz 
| dr daz dr" dy dr‘ dz dr 
135. Let the relations between the variables a, y, z, 6, w, 7, 
be , 


een. OCOs., Y—rsin. Osin. yw, 2=rcos.d (1’). 


From these we find 


ee r cos. 8 cos dy r cos. 6 sin dz r sin. 0 
Eat wie ul Ln = ° e ) - = cas | oe 
do Y Y a9 peed 
da : ‘ d : dz 
dw = — rsin.@ sin. yw, oe =r sin. 0 cos. w, Gib aU, 

dx 3 d , . dz 
ae = sin. 0 cos. y, oe = sin. 7 sin. y, mE ee COS eda 


IP() DIFFERENTIAL CALCULUS. 


and formulas (2), Art. 134, by the substitution of these values, 


become 


oS = 7 cos.6 cos. = + r7cos.6 sin. w a = sind ] 
“ = —rsin.ésin.w os +7 sin. 6 cos. p is (a). 
ae = sin.@ cos. wy ue + sin. sin. w 7 + cos. 6 = 

From Eqs. a maybe found the required values of se i ey, 


du du du 
do’ dw’ dr° 
Again: squaring Kqs. 1’, adding results, and taking square 


in terms of 


root, we have r= /(x*-+ y*+ 27). Adding the squares of 
first and second of these equations, we find r? sin’6=a2*+y’; 


whence 7 sin. 6 = 4/(x? + y’), sin. 6 =i V(e+y"): and from 
this, and the last of Eqs. 1’, we find 


n/ (x? + y*) 


2 1 972 
ony pias 6 tan NY 
v4 


& 


Dividing the second of Eqs. 1’ by the first, we have 
GAT. 4) oe a w= tan. 
Hence we have 
Apt nee 
r= a/(x? + y? + 27), 6 = tan! Vee ye tan 18 (2/). 


From those by differentiation, we have 


Ae ee dr : drt ine 

io oe 6 cos. W, ae a = sin. 0 Sinia 7s Ga cos. 4, 
dl z ee hy cos. 8 cos. w 

eres Ere re y] 
ye A /c8) 1 97? r 


d9 z y __ cos. # sin. w 


y- PLP pe Very? 


CHANGE OF INDEPENDENT VARIABLE. 220 


Coe 4/(e? + y*) sin. 8 


dz wt ttt gt oe 
ay Cl dye cos.w dw 7, 
de — wtty? resin. dy a+y? rsin.0’ dz — 
By the substitution of these values in formulas (1), Art. 134, 
. we have 
du cos.dcos.w du sin. w du . du 
dz r dé rsin.é6 dw ye i 
dus cos.@sin.w du — cos.w du : du 
a 2 fli UB Le bad ee yee b). 
dy r da + r sin. 0 dy Bart Oe dr ) 
du —s sin. 0 du du 
dz re da Rin dr 


dl 
The values bids’ 
to agree with those given directly by formulas (0). 


given by formulas (a), will be found 


EXAMPLES. 
1. Transform 


area (ayN Ay) 
eae + (Ge) — Som ho) 


into its equivalent when neither x nor y is independent, but 
both are functions of a third variable z. 


Substitute for ote and dy their values given in Art. 129 
das” dx ; 

and we have 
dy dx dx dy : dy 
dz? dz ee og is 


Leet ALANA Pom A 
mee da? ie) acs 


and, multiplying through by ( | 


pe kad, Cae eee. dy zs dx 0 32 
” de? da da® da +(4) -5 i(Z)= ee 


222, DIFFERENTIAL CALCULUS. 


If we make «=a, this reduces to the given equation. 
Making y = 2, (2) becomes 
eae 2/ Oa 
Soa —1=0 (8). 
dy Hg) . 
Equation (3) is the equivalent of (1) when the independent 


variable is changed from @ to y. 


2. Change 
3 a* 
(ae) eae 
Xx 


into its equivalent when both w and y are functions of a third 


variable z. 
c(ay\? | (de\" 4, (Py de _ dew dy 
eae) dz + dz “13 dz? dz \, da? da wie 


If y = 2, the above becomes 


sy a 2 ee 
ee ne } dy 


in which y is the independent variable. 
3. Eliminate « between 
bay 


DO 


= 


+ +y=0,anda2?=46 (1), 


and find what the differential equation is when @ is the inde- 
pendent variable, and also when y is the independent variable. 

First suppose both w and y to be functions of a third varia- 
ble, 2; then the differential equation becomes (Art. 129) 


ay dx  d*x dr 1 ch da dx 
8 ite u(g) ta(q) =9 @ 


dz? dz dz? dz ' « dz \dz 
dx dx d6. dy bea 
da do da Gp ee 
dix _4 46 d?x 1 d LIN _1d*6 04 (Z) 


ee iy det od hh ° 92 


a 
From 2? = 46, we have «= 26? 


CHANGE OF INDEPENDENT VARIABLE. yap hs: 


; a oe by their values, we have 


_1 46 d*y 1 4*6 dy dé? dy da 
Meade. ded!” tat ao iG) =9 (9); 


which does not contain w Making y =z, (3) becomes 


d?0 do\? da\3 
eG.) -9(3) pet 


and if, instead, 0 =z, we have 


In (2) replacing x 


dy 
mtg ty=0 


4. Given the relation «=e, to change the independent 


variable, in the differential expression x” 
| By Art. 42, we have 


een Of a" y\ da (| jdty MAY, 
z(? —ieraG a el Gale a 


dA” 
at from « to s. 


— n ood ir Ge tte 

ey hes dic” Bis dart} 

ad d"y dy Ge Pay 
See | ee n — grr . 

Al am 0 a” dig +1 


or, writing the first member in an abbreviated form, 


ad A/G Me BRE 
ee ae ae 


Making » = 1, this gives 


From «= e', we get - —=e'=2; also we have 
dy dy dx a dy . 
Hagieada Us 9). dx 

hence (2) becomes 


224 DIFFERENTIAL CALCULUS. 


When n = 2 in formula (1), then 


and, putting in this the value of x? oe from (3), 


d*y d d dy 
Oe on 1 ee -— i. ieee 
dic? G i) G YS 


The law governing the construction of these equations is ob- 


vious; and we may write, generally, 


ah d d d d d. 


The meaning of the operations denoted in the second member 


d d 
of formula (4) is, that if the expressions ae v ah 


be combined by the rules for multiplication, the result will 
dy 


represent, in terms of indicated differentiations on ds? the value 


Daas 


) 


d”y 


n 
of aw oe . 


5. If we have p= — — , and the relations 


2—=rcos.6, y—rsin.6, find the equivalent for p when a 
change of independent variable is made from «& to 6, and also 


from x to 7. 


When @ is the independent variable, 


CHANGE OF INDEPENDENT VARIABLE. 225 


and, when 7 is independent, 


ech 


d’6 


aS) 

| 

| 
Q 
S 
Ss 


29 


SECTION XIV. 


ELIMINATION OF CONSTANTS AND ARBITRARY FUNCTIONS BY 
DIFFERENTIATION. 


136. WueEn an equation is given in the form 
F(x, y)=c (4), 
the constant c will disappear on the first differentiation, and 
the successive differential equations derived from (1) will be 
identical with those derived from 


F(a,y)=0 (2). 

Though an equation may not be given under the form of (1), 
it often happens that one or more of its constants may be made 
to disappear by successive differentiation alone. 

Let  (y—b)'+(w—a?P—r?=0 (8), 
and differentiate this equation twice, taking x as the independ- 
ent variable. We find 


(yb) +e-a=0 (4), 


yn T+) +1=0 (6); 


and thus the two constants a and 7 of (3) have vanished in 
the two differentiations which lead to (5). A third differen- 
tiation gives 


dty , ody dey _ 
Y= Nima + > ae gat = 9 A 


226 


ELIMINATION OF CONSTANTS AND FUNCTIONS. 227 


From Kqs. 3, 4, 5, and 6, we get 


eve holy. (ela (Oy) 
Go bay dl ba) Oa 
dy d*y 
dy dx dx" _8r%(a—a). 
ee Lee, Sep ote Ss 


and, by eliminating y — b between (5) and (6), we get 


dy\? dy dy /d?y ae 
re) +1} 55 95) Tene 


Between (3) and (4), we may eliminate any one of three con- 
stants a, b, r; and, by taking these constants in succession, we 
should have for our results three differential equations of the 
first order, each containing two of the constants. By a proper 
combination of (3), (4), and (5), we can arrive at two differen- 
tial equations of the second order, each containing but one of 
the constants of the primitive equation; and between (3), (4), 
(5), and (6), we can eliminate all three of the constants, ob- 
taining for the result a single differential equation of the third 
order. 

It thus appears, that, by differentiation and elimination, Hq. 
3 will give rise, 1st, To three differential equations of ‘the 
first order, each involving two of the constants a,b,r; 2d, To 
two differential equations of the second order, each involving 
but one of these constants; 3d, To one differential equation of 
the third order, from which all of the constants have vanished. 

By means of Eqs. 3; 4, 5, the values of a, b, r, may be ex- 
pressed in terms of x, y, and the derivatives of y of the first 


and second orders. Denoting these derivatives by y’, y’”, we 


find 


+1 ACESS PONCHO OF 


228 DIFFERENTIAL CALCULUS. 


137. In general, if we have an equation between a and y, 
and n arbitrary constants, and we differentiate this equation 
m times successively, we shall have, with the primitive equa- 
tion, m+ 1 equations, between which we can eliminate m 
constants. This will lead to a differential equation of the m™ 
order, in which there will be but n — m™ of the constants; and, 
as the constants eliminated may be selected at pleasure, it is 
evident that as many equations of the order m may be formed, 
each containing »—m constants, as we can form combinations 
of n things taken m in a set, which is expressed by 


n(n—1)(n— 2)...(n—m+1) 
OES Ar ] 


When the original equation is differentiated n times, we 
should have altogether n + 1 equations, between which the n 
constants can be eliminated; and, as the resulting equation 
would involve the n™ differential co-efficient of y taken with 
respect to x, it is said to be of the n™ order. The order of 
the highest differential co-efficient entering any of the equa- 
tions at which we arrive,by the steps above indicated, deter- 
mines the order of the differential equation. 

It is worthy of remark, that if any one of the differential 
equations of the m™ order, obtained by eliminating between 
the first m derived equations, and the primitive equation, m of 
the constants entering the latter, be differentiated n — m times 
in succession, then this equation of the m™ order, and its 
nm —m derived equations, would enable us to eliminate the re- 
maining constants; and the final equation at which we should 
arrive would be the same as that obtained by effecting the 
elimination between the primitive equation and its n succes- 
sive derived equations. 


ELIMINATION OF CONSTANTS AND FUNCTIONS. 229 


We et sa 
To illustrate, take the equation a = x — Fae and differ- 


entiate with respect to x. We should find, after reduction, 
yl" (yl +1) —8y'y” = 0, 
which agrees with Eq. 7. 

The theory of the elimination of constants by differentiation 
is sufficiently simple, and needs but little explanation. It is 
referred to here for the reason that a knowledge of the forma- 
tion of differential equations assists in understanding the more 
difficult and highly important operation of passing back from 
such equations to those from which it may be presumed that 
they have been derived. 

138. Functions known and arbitrary may also be elimi- 


nated by differentiation. 


Let y=asin.2; then VT Bs RA So ANY par Sere. 


In 
dy? 
- ° . a 2 me oe a a 0 ° 
an equation which no longer contains the known function 
sin. &. 
Again: suppose s=9(0) in which w and y are independ- 


ent, and g denotes a function of the ratio of these variables, 
the form of which is not given, and is therefore called an arbi- 


trary function. 

a PS SU OL’ salen. 
eee HP Ong HPO, (¢), 
diz dt x 
ea oy! ES Ee ee a 
a0 Ogg =~ 9 O 


230 DIFFERENTIAL CALCULUS. 


This last equation is true, whatever may be the form of the 
x : x Ret 
function oy denoted by ; it may be z =1(5), 2 — Sie ae or 


z—=e,: and for each of these cases the differential equation 
subsists. 

Take the more general case, w = » (v), in which wand v are 
known functions of the independent variables x and y, and of 
the dependent variable z, and g(v) an arbitrary function of v. 
Differentiating «= g(v) first with respect to # and z, and then 


dz 


Rae 
with respect to y and z, and, for brevity, making - eh eH) Ege 


we shall have 


du du ; dv dv 
hk Pig ee © +7) 


du iy eee a dv dv 
yaaa ta cae 
Dividing these equations member by meniber, we have 
du du dv dv 
det? da dat? a; 
du du — dv dv 
dy srl dz dy + as 
Clearing of fractions, and making 


Pp dudv du dv dudv dudv , dudv dudy 


” dy da dz dy “a de de de! a 
we find that the partial differential co-efficients of the first 
order are connected by the equation 

Ppt Gea 
and this equation is in no wise dependent upon the form of the 
function characterized by ; in other words, this function has 
been eliminated. 


139. Suppose c and c, to be two known functions of a, y, Z, 


ELIMINATION OF CONSTANTS AND FUNCTIONS. 231 


expressed by c=/(a, y, %), . =/i(%, y, 2); and that, in the 
equation 
F (x,y, 2, 9(e), 9(%1)) =9 (1), 


9, 1, denote arbitrary functions. Let us see if it be possible 
to pass from (1) to a differential equation which shall not con- 
tain g(c), g,(¢,), or their derivatives. 

The equations 


df di 


Ee es a 2 
Fp ay 8 
ash ba (i hel 
buat Ta? cE ae yar (3), 


that we get by differentiating (1), will contain the unknown 
functions g’ (c), 9:/(¢,), 9” (¢), gy’ (e1), which, with g(c), o,(¢;), 
make six quantities to be eliminated between Eq. 1 and the 
five equations of groups (2) and (3), which are generally in- 
sufficient. Passing to the equations 

Cr a ad? a | 

Gia deidy "day yt SD 
we introduce two additional arbitrary functions g’”(c), g/(¢), 
and only these two. We shall now have ten equations, viz. 
Hq. 1, and those of groups (2), (3), (4), and but eight arbitrary 
functions to eliminate: hence the elimination can be effected, 
and we may have two resulting differential equations of the 


third order. 


We have said, that, in the case supposed above, it is gener- 
ally impossible to effect the desired eliminations without pass- 
ing to Kqs. 4. It may happen, however, that the forms of the 
functions f(x, y, z), f\(x, y, %), are such that Eqs. 1, 2, 3, will 
prove sufficient. 


ger: DIFFERENTIAL CALCULUS. 


Suppose z—g(e«+ay)+ q,(*«—ay); then 


dz 

ny (et ay) + oi (z— ay); 
dz / / 
Pas (x +ay)—ag’(x— ay), 
d*z HW “/ 

qn? (7 + ay) + g,/(x — ay), 
d?z 


Gp © V2 Oy) Oot re 


From the last two of these equations, we find 
d?z As hed 
=, = a" -_-=. 
dy? dic” 

140. Suppose that we have two functions, 


F (a, Te ORLY mi(c)-++) wae Fi (x,y,% C; (ley mi(¢)-+) ao Ui 


in which ¢ is an implicit function of x, y, z, and g(c), g,(¢).-., 
are arbitrary functions of c. Itis proposed by successive dif- 
ferentiations to eliminate c and the arbitrary functions. To 
accomplish this, z and c must be considered as functions of the 
independent variables x, y; then, having differentiated the 
given equations a number of times successively with respect 
to 2, and also with respect to y, we must eliminate the quan- 
tities 
des de. d7¢ id? cimga eG 
° de’ dy’ dx?’ dady Vaya (1); 


g(c), p’(c), p”(c)..-, gic), gi’(e), i’(e).-- (2); 


between the given and the differential equations. 


Let m denote the number of arbitrary functions 


P(e), Pr (C), g.(C).--, 


and m any positive integer; then, if we stop with partial 


ELIMINATION OF CONSTANTS AND FUNCTIONS. 233 


derivatives of cand of g(c), g,(c), of the n™ order, the num- 
ber of terms in series (1) will be expressed by 


eee tee 


The number of the arbitrary functions 


G(), P/(C) ++) P™(C), il), Fr (C) +++) PIMC) ve sy 
will be equal to m(n+ 1). Again: since each of the given 
equations will give rise to two derived equations of the first 
order, three of the second, four of the third, and so on, the 
number of given and derived equations together will be equal 
to (n+1)(n+ 2). Hence to be able, in the general case, to 
eliminate ¢ and its arbitrary functions, and their derivatives 


up to the n™ order, we must have 


(n +1) (n+ 2) 
2 


(n+1)(n+2)> + (m-+1)m,or5-+1>m. 


This condition will be satisfied if m = 2m — 1, which will give 
2m(2m + 1) equations between which to eliminate 4m? + m 
quantities. The number of equations exceeds the number of 
quantities to be eliminated by m: hence there will be, in gen- 
eral, m resulting differential equations. 

When the proposed equations contain but one arbitrary func- 


tion,g(c),of c, they become 


EF’ ( a, Y, %, ©, 7 (c)) = 9, F(a, Y, 4, ¢; 9(c)) = 0, 


each of which gives two partial derived equations of the first 
order; and we shall thus have, including the given equations, 


six equations between the quantities 


dz dc dc 


a fi 
XH, Y, %, p= ae! t= Gy! C, ape dy’ p(c), Q (c), 


the elimination of the last five of which will lead to a single 
30 


234 DIFFERENTIAL CALCULUS. 


partial differential equation of the first order between the 
variables a, y, z, of which x and y are independent. 

If there are but two arbitrary functions g(c), ,(c) of c, we 
should find that the given equations 


F (a, Y, %,C, PC), 7) = 0, F(a, Y, %, C, P(e), ni(e)) = 0, 
with their partial derived equations of the first order, making 
in all twelve equations, would involve twelve quantities to be 
eliminated ; viz., 

i UG. d6y U2 CrtO.C mae 

? dx’ dy’ dx dady’ dy” 

(Cc), 9(c), (Cc), pile), Pile), 91 (Ce): 

hence the elimination cannot be effected, except in special 


cases. Passing to the partial derived equations of the third 
order, we should then have in all twenty equations, with 
eighteen quantities to be eliminated; viz., the twelve above 
given, and 

O70 02c acorns Ree 

dx” da?dy’ dedy” dy” g’’(c), #1 (¢); 


additional: and we may therefore have for our results two par- 


tial differential equations of the third order between a, Yi i 
the latter being the dependent variable. 


In certain cases, it is unnecessary to make as many differen- 
tiations as have been indicated to enable us to effect the de- 
sired eliminations. Suppose, for example, that the given 
equations contain but three arbitrary functions, (c), ,(c), 
g2(c): in this case, m=3, 2m—1=5; and, to effect the 
eliminations, it would be generally necessary to form the de- 
rived equations of the fifth order, and we should have for our 
results three partial differential equations of the fifth order 
between a, y, z. But if the arbitrary functions are so related 


ELIMINATION OF CONSTANTS AND FUNCTIONS. 235 


that g,(c) = g’(c), g2(c) = 9” (c), the proposed equations be- 
come 

F\ «, Y, %, C, p(c), p’(C), p’”(c) = 0, 

F,{ x,y, 2, ¢, 9(c), 9’(c), »”(c) | = 0; 
and these, with their derived equations of the first and second 
orders, make twelve equations, involving the eleven quan- 
tities 

COmOCa G (076d "¢ 
C, da’ dy’ dx” dady’ dy” 
p(c), p’(c), ep” (Cc), gp” (e), (Cc); 
and the elimination will lead to a single partial differential 
equation of the second order. 
If the value c be found, as it may be, theoretically at least, 

from one, say the second, of the equations 
F\a, Y, 2, C, p(C), gi(¢) { = 0, Fe, Y, %, C, p(C), g1(¢) | = 0, 
and this value be substituted in the first, we should have for 
our result an equation of the form 

Fh x, y, 2, (a, y, 2), ¥i(a, y, 2)} =0, 
which is evidently equivalent to the two proposed equations. 
By Art. 139, we shall generally be unable to eliminate the two 
arbitrary functions w, w,, with this equivalent equation and 
its derived equations of the first and second orders; but it 
would be necessary to pass to the third derived equations to 


effect the elimination. 


EXAMPLES. 
1. Eliminate the constant a from the equation 
Wo? 4/1 y? = a (2 — y). 
a/ ey dy 


Ans. Se et 
2 


/1— 2 


236 DIFFERENTIAL CALCULUS. 


2. Eliminate c from the equation 


ee 4? Cas 
dy 
Pi bel foes ory Pi 
Ans. ¥ 2xy Rime 0. 
3. Eliminate the functions e” and cos. x from 
y —e*cos.x = 0. 
d*y ody fs 
Ans. Dot Ie tha 
4. From y = asin.x + bcos. eliminate the functions sin. a, 
COS. &. 
a 
Ans. a + Y= 0. 
5. If y=ce™'*, prove that 
dy 
6. Ify = be” cos. (nx +c), show that 


d 
(1a) 74 — =O: 


d*y dy 2 ond: ogre 
Srrice 2a". + (4 +n*)y=0. 

T. From the equation y = oie ee eliminate the exponen- 
tial functions. 


Ans. y+ 1 0: 


8. From z= q(e*sin.y) eliminate the arbitrary function 
characterized by q. 

Ans, sin. ia — cos. eee 0 

Y lye Y de 

9. From -e ee, ie he —~-—1=0 eliminate the constants 


a; 0,0. 
2 
Ist Ans. wa bt (7) - ae, =a 
d dz\? ag 
2d Ans. yz aity(g)—a= 


ELIMINATION OF CONSTANTS AND FUNCTIONS. 237 


Poser rom wu — af (2) + g(axy) eliminate the functions 


: ig (2), g (xy). 


sue Eliminate the functions from 
u—f(e+y) + ryg(2—y). 
dey d?u d?u d?u iM ( 2u Pale 


2a da? dy?/ 


oo dc! dy dxdy* dy® a«+y 


12. From 
xr we? —y 
s=r(t) ea 8) xv 


eliminate the arbitrary functions /, g, w. 

d*z az dz GE (rae dz. 

2 ee fF |, es gy ae, a ee —_ — == Vi 
Ans. (x carat) - z+ (: oo AG aed dy) 0 


2 2 
13. From the equations phe ee Be pa = w(x, y), elimi- 


nate the variable z ; i.e., change the independent variable from 
% tO &. 
d 
d w(x, Y) te p(a, Y) it 
Ans. 2g (x 7] ) OG hve? ee 
‘ da dvy : 
da? 


14. Eliminate the arbitrary functions from 


e+ hs() 


d*z d* x , a2 dz dz 
Ans. © ag Tt 2axy Sear Sree nay ee ae 


DIFFERENTIAL CALCULUS. 
ea 


PA Rie SiC Oo iNees 


GEOMETRICAL APPLICATIONS. 


SECTION I. 


TANGENTS, NORMALS, SUB-TANGENTS, AND SUB-NORMALS TO PLANE 
CURVES. 

141. The tangent line to a curve at a given point is 
the limiting position of a secant line passing through that point, 
or it is what the secant line becomes when another of its 
points of intersection with the curve unites with the given 
point. It is now proposed to find the form of the equation of 
tangent lines to plane curves. 

Let y =/(x) be the equa- 
tion of the curve RPQ, and 
take on this curve any point, 
as P, of which the co-ordi- 
nates, referred to the rectan- 


‘Ss’ 


x gular co-ordinate axes Oz, Oy, 
are x andy. This point will 
be briefly designated as point 
(x,y). Give to a, taken as the 


independent variable, the in- 


crement Aa, y will receive a corresponding increment Ay, and 
238 


TANGENTS AND NORMALS. 239 


a+ Ax, y+ Ay, are the co-ordinates of a second point, Q, 
on the curve; then, if x,, y,, denote the general or running 
co-ordinates of a straight line passing through P and Q, the 
equation of this line will be 


eta — 
2 22 ele — 


é (x4 wy x), 
or 
A 
Vi Y =<" (@—2). 
Now, conceive the point Q gradually to approach the point P, 
oa 


= will, at the same time, gradually approach its limit 4 ye 


and finally become equal to this limit when Q unites with P; 
but then the secant line becomes the tangent line. Hence the 
equation of the tangent line is 


d 
Wy = 32 (a1 — 2), or yi —Y = Y' (2 — &), 


dy 
dx 
makes with the axis of abscissa. Calling this angle 7, we 


in which is the tangent of the angle that the tangent line 


have 
Dy ae (are ans 9) 
tan em Ua cot "= ay qe 
da 
1 1 
i Sete — 
COS. T Vity? J (% 3? 
1+/(— 
da 
dy 
; y! dic 
; ae a en 
gIn. T Vity? aa} 
1+/(— 
da 


142. The normal line to a curve at any point is the 
straight line passing through the point at right angles to the 
tangent line at that point. 


240 DIFFERENTIAL CALCULUS. 


Since the normal and tangent lines at a given point are per- 
pendicular to each other, denoting the angle that the former 


makes with the axis of « by », we have 


hw laste hacen Lo) Se 

tang: yf = yr 
: ny 
dc 


and, if x,, ¥,, represent the general co-ordinates of the nor- 
mal line, its equation is 
1 —¥= (=), Ory — Y= — Fea). 
y’ dy 


Cor. When the equation of the curve is in the form 


F(x, y) = 9, or the ordinate y is an implicit function of the 


aP 
abscissa, we have (Art. 84) oY — st hence the equation 
dy 


of the tangent line becomes 


(ea) + = 9, 
and that of the normal 
(v7, — )- (ars y= 0. 

145. To find the equation of the tangent line passing 
through a given point out of the curve represented by the 
equation (a, y) = 0, we should make a, y,, in the equation 
of the tangent, equal to the co-ordinates of the given point. 
Then, since the point of tangency is common to curve and 
tangent, the co-ordinates of this point must satisfy both the 
equation of the curve and the equation of the tangent: hence 
these two equations will determine x and y, the co-ordinates 
of the point or points of tangency. In the same way, we may 
find the equation of a normal line passing through a point not 
in the curve. 


TANGENTS AND NORMALS. 241 


Now, if we have two curves, of which the equations are 
F (x,y) =0, F (x,y) —c =0, respectively, the equations of 
the tangent and of the normal to the first curve will be iden- 
tically the same as those of the corresponding lines to the 
second (Art. 142, cor.). Hence, if for given values of a, y;, 
and any assumed value of ¢, the values of x and y be deduced 


from the equations 


dil’ dk 
Py) 0 — 0, Sordi re BA biome py a 


such values will be the co-ordinates of the points of tangency 
of the tangent line drawn through the point (7,,¥,). In like 


manner, the combination of the equations 


(x,y) —c=9, (a, — =) (*1—Y) = 0, 
will determine the points of intersection with the curves of the 
normal lines drawn from the point (x, 7,). 

Since the equation 


dk 
(2-2) +(yi—y) = 0 


is independent of c, it will represent a line which is the geo- 
metrical locus of the points of tangency of the tangent lines 
drawn from the point (#,,¥,), with all the curves which, by 
ascribing different values to c, can be represented by the 
equation F(a, y) —c=0. So also 
(x) — a (2 yo =0 

is the equation of the geometrical locus of the intersections 
of the normal lines drawn through the point (x,, y;) with the 
same curves. Hence, if these geometrical loci be constructed 
from their equations, their intersection with the curve answer- — 


ing to an assigned value of ¢ will be the points common to the 


curve and tangents, or normals, as the case may be. 
31 


2492 DIFFERENTIAL CALCULUS. 


144. Formulas for the distances called the tangent, the 
sub-tangent, the normal, and the sub-normal. 

Def. 1. The tangent referred to either axis of co-ordi- 
nates is that portion of the tangent line to a curve which is 
included between the point of tangency and the axis. 

Def. 2. The sub-tangent is that portion of the axis 
which is included between the intersection of the tangent line 
with the axis and the foot of that ordinate to the axis, which 
is drawn from the point of tangency. 

Def. 3. The normal is the part of the normal line in- 
cluded between the point of tangency and the intersection of 
the normal with the axis. 

Def. 4. The sub-normal is the part of the axis in- 
cluded between the normal and the foot of the ordinate of the 
point of tangency. The relation of sub-normal to normal is 
the same as that of sub-tangent to tangent. 

In the figure, let P be the 


reference to the axis of x, PM 
being the ordinate of P, Pt 
is the tangent, J/¢ the sub-tan- 
gent, PN the normal, and ZN 
the sub-normal. With refer- 
ence to the axis of y, the 
lines of the same name are 


PT, MT, PN’, and UN’, re- 


aE ) 2 OY eee 1 ae dx 
Nowa — an. Pie Migs — tt dy 
dz 


or Mt = subtangent = y a : 


TANGENTS, NORMALS, &c. 243 


Again: 


MN it __ dy 
WP tan. MPN = tan. Pix = rie 


dy 
dx 


Also Pi=—iM + PM =? (F)+y" ease (Ge) +t: 


al. 2 
Pt = tangent = y Ge) +1, 


= a ie ere are 2 2 
and PN = PM + UN =? + r(Z) seh (2) tig 


MN = sub-normal = y 


anh? 
VPN — normal = y JZ) +1. 


Grouping these formulas, we have 
aay ; d 
Tangent = y sl e +1. Sub-tangent = y 7 


d 
Normal = y JZ ZL) +1. Sub-normal = y 2. 
x 


145. A curve may be given analytically by two equations: 
of the form y = g(t), «x = w(t), which, by the elimination of ¢ 
between them, may be reduced to one of the form y= /(z). 
Without. effecting this elimination, the equation of the tangent 
line will be 


dx 
(Yi —Y) ae (1 — a) Os 
and that of the normal, 


d 
(Hy — y) B+ (a — 2) = 


. 


When the co-ordinate axes are oblique, making with each 


other an angle o, the limit of the ratio ot or fa no longer 


244 DIFFERENTIAL CALCULUS. 


sin. T 
sin. (wo —T) 


expresses tan. 7, but In this case, the investi- 


gation and the form of the equation of the tangent line remain 
unchanged; but the equation of the normal line becomes 


eae 


EXAMPLES. 


1. The equation (x, — x)x + (y,— y)y = 0 of the tangent 
line to the circle can be put under the form 


which, if « and y are variable, and x, and y¥, constant, is the 


equation of a circle, the centre of which, having *) : 5 , for its 
co-ordinates, is the middle point of the line drawn from the 


point (x,, y,) to the centre of the given circle. The radius of 
this circle is equal to Va 28 Y, . Now, for assigned values 


of x,, ¥,, the points of contact with the given circumference 
of the tangent lines drawn from the point (x,, y¥,) must be in 
the circumferences of both of the circles; and, since*(1) is in- 
dependent of r, the circumference of the circle of which it is 
the equation is the geometrical locus of all the points of con- 
tact with the given circumference of the tangent lines drawn 
from the point (#,, ¥,) to the different circles that we get by 
causing 7 to vary in the equation x? + y? = r?, 

2. The general equation of lines of the second order (seuie 
sections) 1S 


u— Ay* + Bry + Cx? + Dy 4+ Fx t+ F=0: 


TANGENTS, NORMALS, $c. 245 


du du 

in By + 2Cx +L, Be eee 

and the equation of the tangent is 

(x; —2) (By +2C24+E)+(y —y)(Bu+2dy+D)=0, 
which the given equation reduces to 


w,(By +2Cx+ IL’) + y;(Be+2Ay+D)+Dy+ Lx 4+2F= 0. 


3. The logarithmic curve is that which has y= 7 li 


for its equation. For it we have dy = ant and the equations 
dx. «la 


of its tangent and normal lines are 

ala(y; —y) — (%,— #%) = 0, alae, —@) + (y,—y) = 9. 
The sub-tangent on the axis of y is expressed by ot = = 
and is therefore constant, and equal to the modulus of the 
system of logarithms. 


4. The logarithmic spiral is a curve having 


m2 2 
Bee ty? 
x 


} of tan 12 — —= WW/ a? + y' + y? — UR, 


for its equation; whence 


dx ety? dy «“+y, 
py wy de e—y’ 
and the equations of the tangent and of the normal are 
(%, —a)(@+y)+ (Yi—Yy)(y¥—#) =), 
(,— 2%) (y—#)—(y¥.—y) (@+y) =0. 


When @,, 7, are considered constant, and x, y, are made to 


vary, these last equations represent two circles, the circumfer- 
ences of which cut the spiral in the points of contact of the 
tangents to the spiral which are drawn from the point (x,, y;). 

5. Denoting the tangent by T, sub-tangent by 7,, normal 


246 | DIFFERENTIAL CALCULUS. 


by N, and sub-normal by N,, determine these lines for the fol- 
lowing curves : — 
First, The circle: 2? + y=r’. 


7? ge 


; WV = 7, Ne eee 


T= +7 (r? — 28, {Bytes 


2 
Second, The ellipse or hyperbola: a = - = side 


Cre a0” at )2 
r=\ (Samazg)te-S nas(tx 2), 


2 4 2 
wai Gri)eeeh N = +a, 
a)\a 


Third, The parabola: y? = 2pa. 


T= 2424 ay Die 2a, N=p? (p+ 20), N, = 


The sub-normal in the parabola is constant, and equal to the 
semi-parameter; the sub-tangent is double the abscissa of the 


point of tangency. 


Fourth, The logarithmic curve: #« = i ly. 


alent 2a Ji 221 LN: 221 
= (Gq) +2 =—, Nahe (p+ : 


N, = lae*™. 
In this curve, the sub-tangent on the axis of x is constant, and 
equal to the modulus of the system of logarithms. 

146. The Cycloid is a curve which is generated by a 
point in the circumference of a circle, while the circle is 
rolled on a line tangent to its circumference, and kept con- 
stantly in the same plane. 

Suppose the circle of which C is the centre, and which is 
tangent to the line Ox at the point 0, to roll on this line from 


THE CYCLOID. 247 


O towards x. While the point of contact is passing from O 
to NV, the radius CO, which, at the origin of the motion, was 


perpendicular to Ox, will turn about the centre of the circle 
through the angle VC’P ; and the generating point will move 
from O to P, describing the arc OP of the cycloid. To find 
the equation of this curve, take Ox, Oy, for the co-ordinate 
axes. Let r= CO= radius of the generating circle; o— NC’P 
the variable angle; and «= OR, y = PR, the co-ordinates of 
the point P: then 

£2£=0R=ON—RN= are PN— PQ =reo—r sin.o =r (w— sin. 0), 
y= PR=CN— C'Q=r—r cos. 0 =r (1 — cos. @). 

From y =r (1 — cos. w), we have 
r—y r—y 


1, ~<———_, 
cos.@ = ——, sin. o = + vary — y’, co. = cos. ~! ee 


and these values of , sin. w, substituted in the equation 


x—=r(o— sin. w), give 
cr (cos.— =") =F V/ 2ry —y’, 


which is the equation of the cycloid. The minus sign before 
the radical must be used for points in the are OPO which is 
described while the points in the semi-circumference OLA 
are brought successively in contact with the line Ox; and the 
plus sign must be used for points in the arc O/B. The point 
O’ is called the vertex of the cycloid, or rather the vertex of 
the branch OO’B ; since, by continuing the motion of the gen- 


248 DIFFERENTIAL CALCULUS. 


erating circle on the indefinite line Ox, we should have an 
unlimited number of curves in all respects equal to OO'B. 
From’the equation of the cycloid, we get 


== | ie ees YS 
dy Ore yee ae ee 


Hence the equation of the tangent line at any point is 


n-y= { (0; =o 


and of the normal, 


oo (x; — @). 
If, in this last equation, we make y, = 0, we find 
%,—e#=Vy(2r—y) = rsin.o = RN. 
Substituting the values of al a ; 


for tangent, sub-tangent, normal, and sub-normal (Art. 144), we 


in the general formulas, 


have for the cycloid 


2r y 
Pay | tay |e 


N= V2ry, N= Vy Ora 


which last agrees with what was found above; and from which 


we conclude, that, if supplementary chords be drawn through 
the extremities of the vertical diameter of the generating cir- 
cle in any of its positions and the corresponding point of the 
cycloid, the lower of these chords will be the normal, and the 
upper the tangent, to the cycloid at that point. 


SECTION IL. 


ASYMPTOTES OF PLANE CURVES.—SINGULAR POINTS. —CONCAVITY 
AND CONVEXITY. 


147. Wuen a plane curve is such, that, as the point of 
tangency of a tangent line is moved to a greater and greater 
distance from the origin, the tangent line continually ap- 
proaches coincidence with a certain fixed line, but cannot be 
made actually to coincide with it until at least one of the 
co-ordinates of the point of tangency is made infinite, such 
fixed line is said to be an asymptoée to the curve. Hence 
we may define the asymptote of a curve to be the lmiting 
position of a tangent line when the point of tangency moves 
to an infinite distance from the origin of co-ordinates. 

To establish rules for finding the asymptotes of curves, re- 
sume the general equation of a tangent line 


d 
Piesypias a (1 — x) (Art. 141), 


and find from it the expressions for the distances from the 
origin at which the tangent intersects the co-ordinate axes. 


These are, 
dz 
f= x — y — = distance on axis ofx (1), 


dy 
dy as 
ee ea = distance on axis of y (2). 


Now, there may be two cases in which asymptotes will ex- 


ist: 1st, Both x—y 7 and y — x dy may remain finite for the 


di 


32 249 


250 DIFFERENTIAL CALCULUS. 


values x= 0, y=o. 2d, One of these expressions may re- 
main finite while the other becomes infinite. If the expression 
for the distance on the axis of « is finite while that for the’ 
distance on the axis of y is infinite, the asymptote is parallel 
to the axis of y; and it is parallel to the axis of « when the 
distance on the axis of y is finite, and that on the axis of a is 
infinite. | 
Ex. 1. The equation of the parabola is 
d da 2x 
y= Spe; .°. me a Spr aa oe 


and, for these values, expressions (1) and (2) for z=, y=, 


are both infinite. The parabola, therefore, has no asymptote. 
Ex, 2, The equation of the hyperbola is 


bb) 
a*y? — ba? = — a*b?, ory= at — WV/ x? — a’, 
a 


da bx dz x? — q* a? 
ert odo 0 os 
a 


dy 


which reduces to 0 for v=o: y —2& a will also become zero 
x 
dy b 
when «=o, and dn becomes -— 5 Hence the hyperbola has 
x 
two asymptotes passing through its centre, and making equal 
angles with the transverse axis on opposite sides. 
Ex. 3. The exponential curve: 


d 
eee min, = = a*la, 
da 1 1 
a,” aa i rn 
y—o = a? —aa*la = for « = 00, but = 0 forge 
and, for pte, °Y = orig 
a 


da 


ASYMPTOTES OF PLANE CURVES. 251 


Hence the axis of x is an asymptote to the curve, and ap- 
proaches the curve without limit on the side of x negative. 
In this reasoning, we have supposed a>1. Ifa<1, the axis 
of x is still an asymptote; but, in this case, the curve ap- 
proaches the axis on the side of x positive. 

148. An asymptote to a curve may be defined as the line 
which the curve continually approaches, but which it can 
never meet. An investigation, based on this definition, may 
be given that differs somewhat from the preceding. 

Let y= ax+ 6 be the equation of a straight line, and 
y = ax+ 6+ v the equation of a curve, v being a function of 
a and y, which vanishes when # and-y are made infinite, or, 
at least, when one of these variables is made infinite; then 
the straight line is an asymptote to the curve. Tor the formu- 
la for the perpendicular distance from the point (a, y) to the 


straight line is Bees Y 


Var +1 


point of the curve. Hence when v vanishes, as it does by 


2 hen the point i 
Se wy eh e poin Is a 
V/ a? +1 


hypothesis, for one or both of the values cz =x, y=o, the 
straight line is an asymptote to the curve. 


From the equation y= ax + 6+, we have ogy -|- Pre 
ry x 
whence ois the limit of a when w and y are increased without 


limit. In general, for these values of # and y, takes the 


dy 
ee rao 1 ae 
form 2; but its true value is ah oe = So, also, 8 is the limit 


of y — ax, and @ is the limit of ae therefore, in general, £ is 
et: dy 
fy—— «2. 
the limit of y In” 


When the value of & and # thus determined are substituted 


752, DIFFERENTIAL CALCULUS. 


in the equation y= ax-+, it becomes the equation of an 
asymptote to the curve. 

149. When two. curves are so related that the difference 
of the ordinates answering to the same abscissa converges 
towards zero as the abscissa is increased without limit, or the 
difference of the abscissa answering to the same ordinate 
converges towards zero as the ordinate is increased without 
limit, either curve is said to be an asymptote to the other. 

Suppose we have a curve, the equation of which may be 
made to take the form 


y=axr"+a,x"—'+.-- 1,0. Sp U0): 
then the curve represented by 
y =ax" +a,x"—'4+.---+a,_,e+a, (2) 
will be an asymptote to the first curve. 
So also is that represented by 
y= an fae + fa, eta,F2 ©), 
and 


y =ax" + ayx"-'+--- +a,_\2 +4, +2424 (4). 


It is obvious, also, that of the curves represented by Kas. 
1,2, 3..., any one is an asymptote to all the others. 

Example. Find the asymptotes, rectilinear and curvilinear, 
of the curve represented by 


x*— xy? + ay?—0, or ya | 


xa 
The value of y may be put under the form y = = a{ 1 — Ae 
x 


and, expanding this by the ae Theorem, we have 


yoto(it i+ Stet) ©) 


SINGULAR POINTS. 253 


which expresses the true relation between 2 and y for points 
of the curve far removed from the origin; for then : is less 
than 1, and the series 1 + 5, + eee +.» converges to a fixed 
finite limit. Whence we conclude that the curve has two recti- 
linear asymptotes represented by the equation y= = (2 + 5) 


and an unlimited number of curves, having for their equa- 


tions 


a 3a? a 3a? 
— ies Fae Sos et oo a eos 
Y Pei tot.) é o(l+s toc ) 


which are asymptotes to it and to each other. 

150. Singular points of curves are those points which 
offer some peculiarities inherent in the nature of the curve, 
and independent of the position of the co-ordinate axes. 

First, Conjugate or isolated points are those the 
co-ordinates of which satisfy the equation of the curve, but 
which have no contiguous points in the curve. 

Ex. 1. x?-+ y?=0 can be satisfied only forxa«—0, y =0, 
and represents therefore but a single point; i.e., the origin of 
co-ordinates. 

Pee a (a — @”). Thisis satisfied by x= 0,7 = 0, 
and therefore the origin belongs to the curve: but there are 
no points consecutive to it; for values of « between the limits 
x= +a, «= —a,make y imaginary. Hence the origin is 
an isolated point. 

Ex. 3. ay?— «x + bx? = 0. 

Second, Points d’arrét are those at which the curves 
suddenly stop. | 

Ex. 1. y=alx. Herex=0, y —0, satisfy the equation ; 


254 DIFFERENTIAL CALCULUS. 


but negative values of x make y imaginary. The origin is 
therefore a point d’arrét. 


ix 2a ez. If « be indefi- 
nitely great, and positive or nega- 
tive, y approaches the limit 1; but, 
if x be indefinitely small, and posi- 
tive, y approaches the limit 0; 


while, for negative and very small values of x, y approaches 
+o. The curve will be composed of two branches, as rep- 
resented in the figure, and will have for the common asymptote 
to these the parallel to the axis of x at the distance 1. 

Third, Points saillant are those at which two branches 
of a curve unite and stop, but do not have a common tangent 
at that point. 


Example. y= 


From this we find 


If x be positive, and be dimin- 
ished without limit, both y and a 


ultimately become zero; but if « 


be negative, and be numerically 


A v diminished without limit, we have 
ultimately y= 0, os — 1, -Hence 


the origin is a point of the curve at which two branches unite 
having different tangents; one branch having the axis of x for 
its tangent, and the other a line inclined to the axis of # at an 
angle of 45°. 


SINGULAR POINTS. 256 


Fourth, Points de rebroussement, or cusps, are 
points at which two branches of a curve meet a common tan- 
gent, and stop at that point. The cusp is of the /irst species 
if the two branches lie on opposite sides of the tangent, and 
of the second species if the branches lie on the same side of 
the tangent. 

Fifth, Multiple points are points at which two or more 
branches of a curve meet, but do not all stop, or at which at 


least three branches meet and stop. 


Ex. 1. y?=«x?(1 — x”) represents a curve of two branches 
which cross at the origin, at which the equations of the tan- 
gents arey=— 2%, y= «x. 

Ex. 2..The equation y? = x*(1— 2’) is that of a curve 
composed of two branches which meet at the origin, and have 
the axis of x fora common tangent. The origin is a multiple 
point. 

Sixth, A point of inflexion is one at which the curve 
and its tangent at that point cross each other. 


151, We will now establish the analytical conditions by 
which the existence and nature of singular points in a curve, 
if it have any, may be generally recognized; omitting, for the 
present, the case in which the first differential co-efficient of 
the ordinate of the curve becomes infinite. 

If a curve has either a conjugate point, a point d’arrét, a 
point saillant, or a cusp of the first or second species, we may 
pass through this point an indefinite number of straight lines, 
such that, in the vicinity of this point, there is not on one side 
of any one of these lines for the last three kinds of points just 
named, or on either side for that first named, any point belong- 


ing to the curve under consideration. 


256 DIFFERENTIAL CALCULUS. 


This is illustrated 
in the adjoining fig- 
ure,inwhichJ/,Fig.1, 
is a conjugate point ; 
M, Fig. 2, is a point 
darrét; If, Fig. 3, a 
point saillant; and J, 
Figs. 4 and 5, are 


cusps of the first and 
second species. 

Now, if, for any one of these cases, two points, P, Q, be 
taken on one of these lines, one on each side of the point J, 
and however near to it, these points may be united by a curve 
which has no point in common with the given curve AB. 
Consequently, if w —/(x, y) = 0 is the equation of AB, and u 
is continuous, as is supposed, it cannot change sign, except at 
zero: but no values of x, y, belonging to PQ, can reduce wu to 
zero; for, if so, then that point would be common to 4B and 
PQ. Hence the values of x, y, belonging to points of PQ, 
make the sign of w constant; while the values of a, y, belong- 
ing to the point JZ, reduce w to zero. 

Since, then, the value of w at the point JZ is zero, and has 
the same sign at P, on one side of this point, that it has at Q 
on the other, these points being very near WW, wu must bea 
maximum or minimum at Jf according as the sign of wu at P 
and @ is negative or positive. In either case, we must have 

du _ du dy 
dx dy dx 


Again: denoting the tangent of the angle that the arbitrary 


straight line PJ/Q makes with the axis of x by a, the equation 
of this line, which the co-ordinates of IZ must satisfy, will be 


SINGULAR POINTS. 257 


y =ax-+b: whence ih —a; and, substituting this above, 


we have 


But this last equation must hold for an indefinite number of 
values for a, since the line PIZQ is arbitrary; and therefore 


we must have 


du du 
ean 0, ee 0. 

The co-ordinates of the four kinds of singular points under 
consideration should then satisfy, at the same time, the three 


equations 


Two of these equations will determine values of x and y to 
substitute in the third. Ifa set of these values x=2z,, y= y,, 
verifies the three equations, the corresponding point may be 
a singular point, but not necessarily so. 

To ascertain the nature of the point thus determined, let us 
ce a 
dy ame 


seek the value of a which the equation ae 
gives under the form f The second differential equation, 


because of the conditions au —0 oe 


ee Mee = 0, reduces to 


ete) + 2a ast a! 
dy? \dx dxdy dx. dz 
Suppose, also, that, by the solution of w=—/(a, y) = 0, we 


have found y= (x) for the equation of the branch of the 
33 


258 DIFFERENTIAL CALCULUS. 


curve on which the point about which we are inquiring is sit- 


dy 


uated. The solution of Eq. (a) with respect to op gives 
du 4 d?u\® d*u dra 
iy _ ity \ Gay) 
1 d?u 
rence, dy? 


I. From the definition of a conjugate point and these equa- 
tions, we conclude that the point x = x), y = yy, will be con- 
jugate: first, if the two ordinates 

Y= Fe +4), Y= P(e — A), 
are both imaginary; second, if the curve at this point has no 
tangent, which requires that 
d’'u\? dw da 
(iy) Podat dyke 


unless we have 
hat) Db on Oe 
dat =) dady > “dy? 
Il. The point x=2,, y= yp, will be a point darrét: first, 


=i) 


when only one of the ordinates y = F(x, +h), y= F(a,—h), 
is imaginary ; second if the curve at this point has but one 
tangent, which will be the case when the co-ordinates of the 


: « epee 
point satisfy the equation ae a0. 


III. The point «= x), y¥ = Yo, will be a point saillant: jirst, 
if to each of the abscisse sz =a, +h, «=a, —h, there is but 
one corresponding ordinate, differing but little from y,, or 
if there are two, and but two ordinates, differing but little 
from Yo, corresponding to one of these abscissa, and none to 
the other abscissa; second, if the curve at the point ay, yo, 
has two tangents, which requires that we have 


(cam) du d*u SS 0, 


dady) da* dy? 


SINGULAR POINTS. 259 


IV. The point x, y, will be a cusp, when, the first condi- 
tion for a point saillant being fulfilled, the two tangents at 
that point coincide; which cannot be the case unless 

eu? du du 
(aedy) ost duit 

152. To investigate the conditions for multiple points, let 
the equation /'(x, y) 0 in rational form represent the curve; 
then 


Beene e gS ne) WATE 84), 
da ' dy dx evi 


Since at least two branches of a curve pass through a mul- 


tiple point, two or more tangents may be drawn at that point: 


d 
hence ee) for such a point, must have more than one value. 


dx 
But, since F(a, y) is supposed rational, oe ae will each ad- 
xy 


mit of but one value for the values of x), ¥), which determine 


: dy 
the point. Therefore “! cannot have more than one value, 


dic 

dk dF 
mess = — 0, —— 
dx dy 
existence of a multiple point. The equation from which to 


= 0; and these are the conditions for the 


find the values of dy is 


dlc 
ad? re ct ad?’ /dy\? 
me ay eta.) = © 
dx dxdy dx. dy* \dx 


d 
which will give two real values for A , If, for the values of a, 


and Yq, 
CGE\? @Faer 
(Ger dx? dy? 


and in this case the multiple point is called a double point. 


>; 


260 DIFFERENTIAL CALCULUS. 
CF a’r a’F 

If Soe Se 0, 
dat dady dy” 


then Eq. (b) becomes indeterminate, and we must pass to the 
differential equation of the third order, which, after intro- 


ve BEA a : 
ducing the above conditions, 1.e. Nie 0, Si 0 nuts 
f* aS ad? i’ fdy\?. “ote yag 
isate ge OY so (2 Tale jaa 
a" dx* dy dx dady* \dx dy* \dx 


This cubic equation will give three values for a which, if 


all real, show that three tangents can be drawn to the curve 
at the point (a), ¥)): the point is then called a triple povnt. 
If Eq. (d) becomes indeterminate, we proceed to the differen- 
tial equation of the fourth order, and thus get an equation of 


the fourth degree for finding HY ; and,in general, if n branches 


of a curve unite ina multiple point, the co-ordinates of such 
point must verify the following equations: 


LER OLE se Se ae ak... a 0 
dx’ dy 7 dx? >" dedy ) ‘G75 
thie tO dif ay anes 0: 
dete dx” dy ine dy" ae 
and the n™ differential equation of the curve would in general 


determine 7 real values for th 
x 


153. If a curve has a point of inflexion, the co-ordinates 


2 
of that point must verify the equation ee coy 8 


Suppose the equation of the curve has been put under the 
form y = f(x); then the difference ay of the ordinates corre- 
sponding to the abscisse # and a + h is (Art. 61) 

ite h” 
Ne h(a) +. 13 B(x) fee ote EF (x + oh). 


abe vores 


, 
—- 


SINGULAR POINTS. 261 


The difference of the ordinates corresponding to the same 
abscisse of the tangent line at the point (ax, y) is Ay; =AF"(x): 
hence, denoting Ay — Ay, by 5, we have 

— iaz E(x) + sdf EM (ae) fe eee ich EL (x + oh) 
ey 1.2.3 RR ; 


When # is very small, the first term in the expression for 6 


exceeds the sum of all the others; and consequently the sign 
of 5 for points in the vicinity of the point (x, y) will be con- 
stantly positive, or constantly negative, according as F(z) is 
positive or negative: hence, if #”(x) does not vanish, the 
curve cannot cross the tangent at the point (x, y), and there 
can be no point of inflexion. If F”(x) vanishes, then the first 


3 
term in the value of 6 is sas f(x), if F(x) does not vanish 
-ae v0 


at the same time; and the sign of this term will change from 
positive to negative, or the reverse, as 4 changes from positive 
to negative. This can only be the case when the curve crosses 
the tangent at the point (x, y); and this point is therefore a 
point of inflexion. If £’”(a) = 0, then, by the same course of 
reasoning, we prove that the co-ordinates of a point of inflex- 
ion must verify the equation #’”(x)=0, &c. Thus, to find 
the co-ordinates of a point of inflexion, we seek the roots com- 


mon to the equations 


d*y 
y= F(x), F” (x) =0, or f(a, y) =0, 77 = 0. 
A system x= 2,, y= ¥Y,, of these roots, will be the co-ordi- 
nates of such a point, if the first of the derivatives that does 
not vanish for them is of an odd order. 
154. Throughout this investigation of the conditions for 
singular points, we have supposed /(x), and its derivatives 


for values of x and y in the vicinity of those corresponding to 


262 DIFFERENTIAL CALCULUS. 


: go 
the point (,, y,), to be continuous. But, if oe co, We may 


dx 
readily determine the nature of the point (x,, y,). Under 
this hypothesis, the two quantities #'(x, +h), F(a, —h), may. 
both be real; or one may be real, and the other imaginary. 

First, If both are real, and both greater or both less than 
F'(x,), the point (x,, y)) will be a cusp of the first species: if 
one is greater and the other less than /’(@,), the point will be 
a point of inflexion. 

Second, If one of these quantities, say F(x, — h), is real, 
and the other imaginary, then, if #’(~, — h) has but one value, 
the point will be a point d’arrét: if (a2, — h) has two values, 
both of which are greater or both less than /(z,), the point 
will be a cusp of the second species; but, if one of these values 
is greater and the other less than /(z,), the point will be 
simply a limit of the curve. 

Third, If each, or but one, of the quantities 

F(a, +h), F(2,—h), 
has more than two values, the point (x,, y,) will be, in gen- 
eral, both a multiple point and a point of inflexion. — 

In conclusion, to obtain the co-ordinates of singular points 


of curves, we seek the values of x and y that will reduce the 
: Rs 0 
differential co-efficients to zero, to infinity, or to x The na- 


ture of the point is ascertained by inquiring how many 
branches of the curve pass through the point, and determin- 
ing the position of the tangent line or tangent lines corre- 
sponding to the point. 

155, The terms “concave” and “convex” are employed 
to express the sense or direction in which, starting from a 
given point, the curve bends with reference to a given line 


CONCAVITY AND. CONVEXITY. 263 


from the tangent at that point. If it bends from the tangent 
towards the line, it is said to be concave, or to have its con- 
cavity turned towards the line; but, if the sense in which it 
. bends from the tangent is from the line, it is said to be convex, 
or to have its convexity turned towards the line. 

To find the conditions of the concavity or convexity of a 
curve towards a given line, take that line for the axis of a, 
and let P, of which the co-ordinates are 2 and y, be the point 
at which the curve is to be examined with reference to these 
properties. Draw the 
tangent at P: then, 
from our definition, if 
at P the curve be con- 
vex to the axis of 2, 
the ordinates of the 
curve for the absciss 
2 +h, « —h, must be 
greater than the corresponding ordinates of the tangent at P; 


h having any value between some small but finite limit and 
zero. But, if the curve be concave towards the axis of a, the 


reverse must be the case. 


If the equation of the curve is y = F(x), the ordinate cor- 
responding to the abscissa « + h is 


yt by=F (x) +hF"(x) + an FY (g) to 


HM 


he 


ig ay, ne Coste eee 


The equation of the tangent to the curve at the point (a, y) 
is yj — ¥ = F' (x) (a, —2), or yy = F(x) + 0, LF" (x) —a2 l(a). 
Observing that x, y, are the co-ordinates of the point of tan- | 


264 DIFFERENTIAL CALCULUS. 


gency, the ordinate of the tangent corresponding to the ab- 
scissa x + h is . 
Y, + AY, = F(a) + oF" (x) + AF" (x) — ck" (a) 
= E(x) + hk’ (x): 
hence, if 6 denote the difference y + ay—(y,+4y;,), we 


have 


h? hn 
— __ fev eye Be 
Be iio Patt Umi nen 


The sign of this difference, when h is very small, is the same 


F(a + 6h). 


72 
as that of 3 which has the sign of #”(x) whether 


h be positive or negative: therefore, if /’”(x) be positive, the 
curve is convex to the axis of ~; and it is concave if F”(a) be 
negative. | 
We have supposed the point of the curve at which its con- 
vexity or concavity was examined to be above the axis of a, 
or to have a positive ordinate. Had the point been below the 
axis, /"”(x) positive would have indicated concavity, and 
i” (x) negative would have indicated convexity. To include 


both cases in one enunciation, we say, ‘‘ When a curve at any 


2 


Yas 


point is convex to the axis of a is positive at that 


2 
point ; when it is concave to the axis of a, y — is negative.” 
1a 
Cor. Comparing this article with Art. 153, we conclude, that, 
when a curve has a point of inflexion, it will be convex to a 
given line on one side of the point of inflexion, and concave 


on the other. 
| EXAMPLES. 


Find the asymptotes to the curves represented by the fol- 


lowing equations : — 


EXAMPLES. 265 


fey = "(2a — 2). Ans. y¥ = — oh oe 

ME y? = (x — a)? («@ —C). Ans. y= x*— $(2a+¢). 

poe y2— a(x? — y*). ATS yf Saeed, 
A, (y — 2x) (y*? — x?) —a(y—ax)+ 4a?(x@+ y) =a’. 

NST) Yi Wy pes ope ee Tm es 

3 3 


Find and describe the singular points in the curves of which 
the following are the equations : — 


3 
x ° e . . e e 
ede aay There is a point of inflexion at the origin, 


and also at the point having # = + a4/3 for its abscissa. 

6. y(at — b*) = x(x—a)*—ab*. There are two points of 
inflexion corresponding to the abscisse «=a, «= = 

T. y3=(e%—a)(x—c). There is a cusp of the first spe- 
cies at the point of which « = a is the abscissa. 

8. «a! — ax?y— aty?+a’?y?=0. There is a conjugate 
point at the origin. 

9. ay? «*?+ bx? =0. There is a conjugate point at the 


4b 


origin, and a point of inflexion at the point having «= z for 


its abscissa. 
84 


SECTION Il. 


POLAR CO-ORDINATES. — DIFFERENTIAL CO-EFFICIENTS OF THE ARCS 
AND AREAS OF PLANE CURVES. — OF SOLIDS AND SURFACES OF 
REVOLUTION. 

156. Let the pole coincide with the origin of a system of 
rectangular co-ordinate axes: denote the radius vector by r, 
and the angle, called vectorial angle, that it makes with the 
axis of « taken as the initial line, or polar axis, by 0; then 
the formulas by which an equation expressed in terms of rec- 
tangular co-ordinates may be transformed into one expressed 
in terms of polar co-ordinates are x =r cos.6, y=rsin. 0. 

To express in polar co-ordinates the tangent of the angle 


that a tangent line to a curve makes with the axis of x, we 


have, calling this angle 7, tan.c= a and hence (qs. a, 
x 
Art. 132) : | ap 
sin. sae + rcos. 0 
tails (Sa ay ees 


dr i 
cos. === 7 8insw 


dé 

and from this we may readily find the expression for the tan- 
gent of the angle that the tan- 
gent line at any point makes 
with the radius vector of that 
point. 

Let MZ be the point, P the 
pole, ZT the tangent line, and 
Px the axis of x, from which 6 


is estimated ; then 


266 


POLAR CO-ORDINATES. 267 


PMT = MTx — MPT: 
hence, by the formula for the tangent of the difference of 


two arcs, 
in. 0 Hh 0 
sin. 6 — Y COS. 
a0 au Sean Pals) 
hate yee 

e feet NS) ° 
cos Wp ry sin do 
tan. PUT = St ee 


ta o( emt ) 
Le SL tanen TCOS. 
sas Neo CUR GH Later 


dr d 
cos.d— — 7 sin. 0 


da 
This may also be found directly as follows: Take on the curve 
a second point, Q, the co-ordinates of which are 7 + ar, 6+ a9, 
and draw JZN perpendicular to PY; then ZN —r sin. 40, and 
QN=r-+ar—vrcos. ad: hence 
fon) VOU r sin. AO 
r+ Ar — rcos. Ad 


Now let the point Q move towards JZ. The limiting position of 

the secant QM is the tangent J/7’, and the limit of the angle 

NQM is the angle PMT. Call this angle 6; then | 
r sin. Ad r sin. AM 


tan. 8 = lim. =i lim, 
r+ Ar — r Cos. Ad 


ee Pea, 
2r sin. 3 + ar 


r sin. AQ 
= lim. Ad 
2r sin. ? — 
Ar 
AO 1 AO 
sin.” ae sin 
: in.? — — 
re sin. AO : : ’ AO 
The limit of —— — 1, lim. adaae oP ain eae 0, 
AG 2 
Ar dr do 
‘lin. —is aenoted by —: therefore tan.B— 7 —- 
aaa ie y do re 


268 DIFFERENTIAL CALCULUS. 


15%. To find the polar equations of the tangent and nor- 
mal lines to a curve, we may assume the equations of these 
lines referred to rectangular axes (Arts. 141, 142), and change 
them into their equivalents in polar co-ordinates; or we may 


proceed thus : — 


Let r and @ be the co-ordi- 
nates of the point M7; and r’, 6’, 
those of a second point, Z, in 
the tangent line: then from the 
triangle PLY, making 


PMR 
we have 
whe sin. PLM _ sin. (6 — 6’ +7) 
7 STD oh sin. T 
= sin. (6 — 6’) cot.t + cos. (0 — 0’), 
or “= 3 au sin. (6 — 0’) +-,cos. (9 — 0’). (CE) 
observing that cot.7 = iat 2 =i 9 (Art. 106). Eg. 1 may 
tan. T 
be written, 
Poe oy a sin.(@— 0’) (2). 
1 1 1dr edd 
Mak —— pe Soka —— = : 
STARS ae eC Mtoe tear then 3 1p ae and hence, by 


dividing both members of (1) by 7, and substituting these val- 


ues, we find 
u’ =u cos. (0 — 0’) — Beat sin. (@— 0’) (3). 


To find the polar equation of the normal at any point of a 
curve, denote by r and @ the co-ordinates of IZ; and by r’, 6’, 
those of any point, #, in the normal: then 


POLAR CO-ORDINATES. 269 . 


: / A 
ie ein RM (0 eae :) 
PR sin. PMR cia ( ; 


7 


7 = sin. (6’ — 6) tan. t + cos. (6’ — 8) 


therefore 


= sin. (0/ — 0) Gas cos.(6’— 6) (4), 


which may be written 
d dr 
() rie Les Vb emer See 
Drage COR (8 fad iniiea: 
Adopting a notation like that in the case of the tangent, (4) 


becomes 


dp 
u’ = u cos. (0/ — 6) — ee) sin. (0’— 6) (5). 


158. Let P be the polar point to which is referred the 
curve RMS, and through P draw 
NT perpendicular to the radius 
vector PM; then MT being the 
tangent, and JZN the normal, to 
the curve at the point J/, the lines 
MT, PT, MN, and PN, are, re- 
spectively, the polar tangent, sub- 


tangent, normal, and sub-normal. 


To find the formulas for the lengths of these lines, put angle 


PMT = §, and resume the equation tan. § = ro (Art. 156), 
r 


Aces Ak : 
‘making oe whence tan. 6 = ets from which we find 


ie Tees 
VS ip Vp py 
Then the triangle Pi re gives 


cos. Bp = 


270 DIFFERENTIAL CALCULUS. 


MP "Sees 
— ee ee ee a oo (74 vy ies 
Ee cos. PMT cos. B Sein cei “ =r Jitr a, ’ 
PTT = PM iP Ma +o = egiee 
74 dr’ 
PM PM __.. ee 
) beh: ire ker, Lito ie 2 pe 2 ig 
Als cos. PUN sin. PMT Ve =.) +(3) ? 


dr 

do | 
The polar sub-tangent is considered positive when it is on 

the right, and negative when on the left, of the line PM; the 


eye being supposed at P, and looking from P toward J. The 
dr 


sign of the sub-tangent will then be the same as that of We? 


that is, positive when 7 is an increasing function of 6, and neg- 


PN=N, = PMtan. PMN =r cot.g =r’ =r’ = + 
r 


ative when r is a decreasing function of 6. 


159. An asymptote to a curve referred to polar co-ordi- 
nates is a tangent line, the polar sub-tangent to which remains 
finite when the radius vector of the point of tangency becomes 
infinite. Hence, to find the asymptotes to a polar curve, we 


must seek the values of 6, which make r infinite while ae 


remains finite. If m be a value of 6 which satisfies these con- 
ditions, the asymptote may be constructed by drawing through 
the pole a line, making, with the initial line, the angle m, and 
another line at right angles to this through the same point; 
laying off on the latter, to the right or to the left according as _ 

do 


Y ee is positive or negative, the distance represented by yr? ay? 


and through the extremity of this distance drawing a line par- 
allel to the first line. The line last drawn will be the asymp- 
tote. 


POLAR CO-ORDINATES. ree | 


Example. In the hyperbolic spiral, so called because of the 
similarity of its equation r = - or r9 =a, to that of the hy- 
perbola referred to its asymptotes, we have 

Boy 08) te coy va 9h 
i UE alt) mC a eet Oe 


Hence the sub-tangent is constant, and equal to —a: but 


Se Oe 


6=0 gives r=; whence the line parallel to the polar axis at 
the distance from it equal to — a is an asymptote to the curve. 

This curve, beginning at 
an infinite distance, contin- 
ually approaches the pole, 
making an indefinite num- 
ber of turns around without eo 
ever reaching it. 

160. When the curve in the vicinity of a tangent line at 
any point, and the pole, lie on the same side of the tangent, 
the curve at that point is concave to the pole; but, if the 
curve and the pole lie on opposite sides of the tangent, the 
curve in the vicinity of the point of tangency is convex to 
the pole. 

Let Pp =p be a perpendicular 
from the pole on the tangent to the 
curve at the point (6,7): then it 
is plain, that, if the curve at this 
point is concave to the pole, p will 
increase or decrease as r increases 


or decreases; that is, p is an in- 


creasing function of 7; and, on 
the contrary, if the curve is convex to the pole, p is a decreas- 
ing function of 7. Hence, when the curve is concave to the 


272 DIFFERENTIAL CALCULUS. 


pole, a must be positive ; and, when convex, a must be neg- 
ative (Art. 51). 

Cor. At a point of inflexion, the curve with reference to the 
pole must change from concave to convex, or the reverse. 


oe ash! : 
Hence, for a point of inflexion = 0 one 


> dr 
We have (Art. 158) 
: “ r chs Lr ; 
Siar i I Vries = Jr+® Tair 
+) 
art raed c r? Le uh dr\? 
a pas pT ee aii 
r+(q) 

i du «1 dr | (dr ae 
Make eer then aca Sa eaae (5) —-,r G; 
1 du\? 1 dp d*u\ du 
BE ey ae ay —— es 
p iat Gi 1 And p*® do (ut 7m) do’ 

dp do _dp_ ( au 

do du du P wt oe) 

dp _dpdu____ 1 dp_p’ Bu 
Bence dr dudr rdu_ r* a im) 


2 


_ will generally change 


Therefore, at a point of inflexion, w + 
its sign. 
161. Differential co-efficient of 
s the arc of a plane curve. 
Let: F(a, y) = 0 or Yaya) 
be the equation of the curve 


RPS referred to the rectangular 
axes Ox, Oy; and take, in this 
curve, any point, P, of which the 
co-ordinates are x,y. Denote the length of the curve estimated 


DIFFERENTIAL CO-EFFICIENTS OF ARCS. 273 


from a fixed point to the point P by s; then, if x be increased 
by MM’ = Aa, s is increased by the arc PP’ = As, and it is 


required to find the limit of the ratio = , or the differential 


co-efficient of the arc s, regarded as a function of &. 

The tangent line to the curve at the point P meets the or- 
dinate P/M’, produced, if necessary, at Y ; and makes, with the 
axis of x, an angle of which the tangent, sine, and cosine are 
respectively 

/ y' 1 
OS Ey Eg 

Now, if, within the interval az, the curve is continually con- 
cave or continually convex to the chord PP’, it is evident 
thaware sf’ >‘chord PP’, and arc PP’ << PQ + QP". 
But chord PP’—/ az? + ay?, PQ= soa, OPN pT 

ieee NV tans OPN = ylAme oe QPimy' sg —Ays 
hence, substituting in the preceding inequalities, we have 

As > Waa? + ay?, as <AaV 1+ y? 4+ yaa — Ay. 
Therefore 


AS Ay? a 
Ae Bee ie oT ek rae 


At the limit, the second member of each of these eye 


peeee gia 
reduces to W1 + y” = J + e) : hence we have 


As , ,. fay? 
lim. eimai = i+(f 2 


for the differential co-efficient of the arc regarded as a func- 
tion of the abscissa. This must be understood as expressing 


only the absolute value of 2 


; for the arc may be an increas- 


35 


974 DIFFERENTIAL CALCULUS. 


ing or a decreasing function of the abscissa, according as it is 
estimated in the direction of x positive or a negative: hence 
the general value should be written 


ds PTE dy\? 
a V1 = ey © 
ae VI Eg + le 


c 
the sign + to be taken if s increases with a, and the sign — 


in the opposite case. 


Cor. 1. Since 
tim, pO OP! iy SOV ee 
PEE VJ Ax? + ay? 


Rests A 
V1+y’ ty 

é 2 
ih ——————————————— ll, 


A 2 

Nt+(@) 
and the arc PP’ is always included between the chord PP’ and 
AS ae 


and hence, when the arc is infinitely small, it and its chord 


it 


the broken line PQ+ QP’, it follows that lim. 


become equal. 
Cor. 2. Squaring both members of the equation 


ds dy\? ) 
eae eG g 


cy oe da\? 
and multiplying through by (=) , we find 


Cation ag 
= a 
Now, if # and y are both functions of a third variable z, the 
P dx ey 
dame tee os 1d ae 
Gagade de.” ds. a; idem 
dz 


DIFFERENTIAL CO-EFFICIENTS OF ARCS. 275 


dy _dyds| . dy dz 
dei dsedgiee © os 


These values of ee Be substituted in the preceding equa- 


a {y+ 


162. Denote by a and f the angles that the tangent line at 


tion, give 


the point P (figure of last article) makes with the axes of « 
and y positive; then 


; Ax 1 dx 
eg ne (Arts. 41, 161), 
WV Ax? + ay? Vi+y! ds 
Ay _dyduz_ dy 


GOaap) =’ lim. 


ESS Se — A 17 49, . 
Vaxttay? dzds ds euinak. 

or, more generally, writing the sign + before Waa? +ay?, 
dy 


ale 
Gos, C= Tee cos. B = Am 


in which the upper sign or the lower sign is to be used ac- 
cording as the angle is that made with the axis by the tangent 
produced from the point P in the direction in which the arc 
increases from that point, or the opposite. This refers to the 
algebraic signs of cos. @, cos. f: their essential signs are deter- 
mined by combining their algebraic signs with the essential 
a dy 
es 


168. Differential co-efficient of an arc referred to polar 


signs of 


co-ordinates. 
For the transformation of rectangular into polar co-ordinates, 


276 DIFFERENTIAL CALCULUS. 


we have « =rcos.6, y=—rsin.6. We also have (Arts. 42, 


161) 
G 2D aN ey dy\? 
do” dx do” do 1+ (3) = (a) + Ge) 
dx r d “+ or 
But — peasy amma 6, —~ = sin. 0 + 1oos. 0; 
ap? 


ds Bex: 
therefore orm re =) : and, in like manner, 


dr ib 
dr dédr — ag () 


Cor. When £ is the angle included between the radius vec- 
tor of a curve at the point (7, 0) and the tangent line at that 


dé 
point, we have (Art. 156) tan. 6 = r — ; and hence 


dr’ 
dé do 
dr _ ‘dr do 
“yn sea aes ds’ 
dr dr 
and ui 1 dr 


te ey eam 
dr dr 

164. Differential co-efficient of the area of a plane curve. 

The area enclosed by the arc 
RP of a plane curve, a given 
ordinate AR, the ordinate of any 
point P of the curve, and the axis 
Ox,is obviously a function of the 
abscissa of P, since the area va- 


ries with the position of this point 
on the curve. | 
Let ARPM =u, and give to x, the abscissa of P,the increment 


DIFFERENTIAL CO-EFFICIENTS OF AREAS. 277 


MM’ = ax; then MPP’M’ = awis the corresponding incre- 


Fae By! 
ment of wv, and we are to find the limit of the ratio ce 
a 


Through P and P’ draw parallels to Ox, and limited by the 
ordinates PM, P/M’; and suppose Az to be so small, that, be- 
tween these ordinates, the ordinate of the curve constantly 
increases or constantly decreases. The rectilinear areas 
MPP'M’, MN’ P’M', have yax, (y+ Ay) Aa, for their respective 
measures, and the curvilinear area PP’ is constantly in- 
cluded between these two; that is, 

Au > YyAx, AU<(y+tAy)Aa: 
whence y < — <y-+ Ay; or, by passing to the limit, 
= ae (i ae; 

If the co-ordinate axes are oblique, making with each other 
the angle w, the above demonstration still applies, observing 
that then the area Aw lies between the areas of two parallelo- 
grams, the sides of which are parallel to the axes; and, since 
the area of a parallelogram is measured by the product of its 
adjacent sides multiplied by the sine of the included angle, we 


d 
should have “ = ¥ SID. o. 


165. When the curve is re- 
ferred to polar co-ordinates, the 
area considered is a sector em- 
braced by a given radius vec- 
tor Pf, the radius vector PM of 
any point (6,7), and the are RM. 
Denote this area, which is a func- 


tion of 9, by u; let 6 be increased 
by 40, by which the point moves to S, and wu is increased by 


278 DIFFERENTIAL CALCULUS. 


the sector PMS=a wu; and suppose Aé so small, that the radius 
vector of the arc from J/to S is constantly increasing or con- 
stantly decreasing. With P as a centre, and the radii vectores 
PM=r, PS=r',as radii, describe the arcs J, SK, limited 
by these radii vectores. We have 

sector PMI < Au < sector PSK; 


1 1 
or, since sector PMJ == 3” 26, and sector PSK = = p48, 


1 /2 1 ad 2 
aro < Aw <s Sy Paks ais ey? < ae 
But, at the limit, 7’ becomes equal to 7: hence 
Cs Mil 1 
aa ae TOS ie gr 0. 


The equations «=rcos.6, y=rsin.6, give J — tan.0: whence, 
a 


by differentiating with respect to 8, 
ot de. ''e4) | oridosaiam 2 
“do 7 do cos20 costo.” 


5 (ody — ydx) = srtdd; 


an expression in terms of rectangular co-ordinates for the dif 
ferential of a polar area that is of frequent use. 

166. Differential co-efficient ue the volume of a solid of 
revolution. 

If V represent the volume 
generated by the revolution of 
the plane area ARPM about the 
axis Ox, and x be increased by 
MM’ = aa, the corresponding in- 
crement AV of V will be the vol- 
ume generated in the revolution 
| by the area MPP’M’. Now, if 
Ag be so small, that y increases constantly from P to P’, AV will 


DIFFERENTIAL CO-EFFICIENTS OF VOLUMES. 279 


be included between the volumes of the cylinders generated 


by the rectangles | 
MPNM, MN'P'M. 


Hence, denoting MP by y, and M’P’ by y,, we have 
AV 
my? ax < AV < my’ Aa, or my? < e < my? 
that is, aa is comprised between the two quantities my’, 2y?, 


the second of which converges to equality with the first as 
Aw diminishes. Hence, at the limit, 
oe = ay’, dV = ny? da. 

167. Differential co-efficient of the surface of a solid of 
revolution. 

Let s represent the arc RP (figure of last article), and 8’ 
the surface generated by the revolution of this arc about the 
axis Ox. For the increment MMW = ax of a, s will be increased 
by the arc PP’ = 4s; and S, by 4S = the surface generated 
by 4s. When Az is sufficiently small, the surface aS will be 
comprised between the surface of the conical frustum gener- 
ated by the chord PP’, and the surface generated by the broken 
line PQP’. The surfaces generated by the chord and by the 
broken line are measured by 


m(2y + ay)V/ aa? + ay’, 


9 dy ee 9 dy Ci ees 
mu eee V (ac) + (Ay)? + een Sune 

— Inyay — n(Ay)*: 
hence, establishing the inequalities, and dividing through by 


Aw, we have 


280 DIFFERENTIAL CALCULUS. 


S 2 
Se > n(2y + ay) [1+ fe 
Be a(2y +9042) + (5 Ly) a(t Nan —n(2) ax 


dy ae 
a ——— © 
+ Qry 7 ny 


At the limit, the second member of each of these inequalities 


dy? 
becomes equal to 27y J oo ey : hence 


TAI Ta at i dy ds 
Din an aed 1+ (FE) = aay 2 


dS = 2nyds. 


SECTION IV. 


DIFFERENT ORDERS OF CONTACT OF PLANE CURVES. — OSCULA- 
TORY CURVES. — OSCULATORY CIRCLE.— RADIUS OF CURVA- 
TURE. —EVOLUTES, INVOLUTES, AND ENVELOPES. 


168. Suppose y = F(x), y=/(x), to be the equations of 
the two curves RPN, h’PN’, 
which have a common point P; 
and let us compare the ordi- 
nates M’N, M’N’, of these 
curves corresponding to the 
same abscissa OM’ —ax-+h, 
differing but little from the 
abscissa OM = «x of the point P. 
We have 


MN=F(ath) WN=f(x +h): 
NN = F(x +h) — f(a +h). 


Developing each term in the value of NN’ by the formula of 
Art. 61, observing that F(a) =/f(«) by hypothesis, we find 


mech. df\., i (Br, dif 
MW = i) + cages — an) + 


h” (ae ZZ) Rett (a it) 


da” dx” 


1:2 1.2..n+1 dat) dan+! 


hr? 


Ree ee (Fe +(x + oh) —f"t?(x + 1h); 


the last term of which may be written, — 
36 281 


282 DIFFERENTIAL CALCULUS. 


n-+1 
oan pa (Bete +m) —p4(@+ 02) 
hrri R 
12s 


# being a quantity that vanishes with 4: hence 
2 2 
NN’ —} dk df ee ar a‘f 
dz dx dx* dx? 


Arr qd? +ift Chass 
et (ae dx” +} ): 


dls gd 
If, in addition to F(x) = f(x), we have Fae es —S the curves 
x 


have a common tangent, PZ’ at the point P, and are a to 
have a contact of the jirst order: and if, at the same time, 
PP _d' 
Og? da 
the contact is of the n™ order if n denotes the highest order 


4, the contact is of the second order; and, generally, 


of the differential co-efficients of the ordinates of the two 
curves that become equal when in them the co-ordinates of the 


common point are substituted. 


169. When two curves have a contact of the n™ order, no 
third curve can pass between them in the vicinity of their 
common point, unless it have, with each of the two curves, a 
contact of an order at least equal to the n™. For y= F(z), 
y —f(x), being the equations of two curves, APN, LPN’ 
(figure of the last article), which have at the point P a contact 
of the n™ order, let y = g(x) be the equation of a third curve, 
R’ PN”, passing through P, and having with the first curve a 
contact of the m™ order, m being less than n. Then, by the 
preceding article, we should have © 


dF dg A” i See 
E(x) = OO eee da™ Pian 


CONTACT, OSCULATION, §c. 283 


nee ym d™+iff duty 

ee NN 1.2...(m +1) eae dnt 
pnt qztipt . qe+ 

i Ree a an . a ee) (); 
FR and F#, being quantities which vanish with h: hence 

Galen vm ie" eae 

ee er ae ae 

WN (i 2).-.(n + 1) gatip  datig 

dam™+i — qym+ 


+1) (a), 


also. cAV4— 


+ £, 


Since, as 2 converges towards the limit 0, & and &, converge 


towards the same limit, and reach it at the same time h does, 


; ‘ NN 
and since n > m, it follows that the ratio ——— can be made 


NN” 
as small as we please by giving to / a sufficiently small value; 
that is, when / is a very small quantity, VV’ will be less than 
NN”, and the curve y= q(x) cannot, in the vicinity of the 
common point P, pass between the curves y = F(z), y = f(z). 

It is evident that this reasoning holds when h is negative as 
well as when it is positive. 

Cor. When h is sufficiently small, the sign of the expression 
for VN’ (Eq. 6) will be the same as that of A”+}, and will 
therefore change with that of 4 if n be even, but remain in- 
variable if n be odd. Hence, if two curves have a contact of 
an even order, they will cross each other at the point of con- 
tact, but not otherwise. 

170. Osculatory curves. If the form of the function 
F(x), and the constants which enter it, are given, the equation 
y = F(x) represents a curve fully determined in respect to 
species, magnitude, and position; but if the form of the func- 
tion only is known, the constants which enter it being arbitrary, 
the species of curve is all that the equation determines. Thus 


the equation y = b+ Wr? — (a — a)’, when 7, a, b, are fixed in 


284 DIFFERENTIAL CALCULUS. 


value, represents a circle that is completely known; but if 7, a, 
and 6 are undetermined, the equation may represent every 
possible circle lying in the plane of the co-ordinate axes. 
It is then the equation of the species “ circle.” 

When a curve of a given species has a higher order of con- 
tact than any other curve of that species with a given curve, 
the former is said to be an osculatrix to the latter. 

Suppose /(x,, Yi, a, b,c...) =0 (1), involving +1 arbi- 
trary constants, to be the equation of the species of curve that 
is to be made an osculatrix to the curve of which y= F(z) (2) 
is the equation. By means of the n+ 1 constants in (1), we 
can satisfy the n + 1 equations 

dy dy, dy dy, dry dry, 
ATI Fy ide) det = dei. ae (3); 

or, in other words, these equations will determine the values 
of a, b, ¢..., which, substituted in (1), will make it the equa- 
tion of a curve having a contact of.the n™ order with the 
curve represented by (2); and it will be an osculatrix, since, 
in general, a higher order of contact cannot be imposed. We 
conclude from the above, that the number denoting the order 
of contact of an osculatory curve is one less than the number 
of constants entering the equation of the curve. 

Example. The form of the equation of a straight line is 
y—ax-+b; and since this equation contains two constants, 
a and b, we may so determine them as to cause the line to 
have a contact of the first order with a given curve at a given 
point. Suppose y= f(x) to be the equation of the curve, 
and that «=m, y = n, are the co-ordinates of the point; then 
the equations to be satisfied are 

am+b=F(m), a= F’(m), 
which determine a and 0. 


OSCULATORY CIRCLE. 285 


171. Osculatory circle, or circle of curvature. 
Assume the co-ordinate axes to be rectangular, and let 
y = F(a) (1) be the equation of the given curve; then, since 
(x, —a)?+ (y,;— 6)? =p? (2) is the general equation of the 
circle, and contains three constants, the osculatory circle will 
have, with the given curve, a contact of the second order. 

From (2) we get, by two successive differentiations, 


da 
Pie Canty 1 9) a = 


14 (H+ n 


and, because the circle is to be ie circle of curvature, we 


(3); 
—0 


must have 
dy dy, dy d’y, 
Y= 913 dx dx,’ dah Ga oo 
dy, dy 
day’ Fibs 7 


e—a+ (yb) =0, +(2 ZL) +u-9 G4 =0 (5): 


These values of 7, substituted in Kqs. 3, give 


therefore 
dy dy 4 dy 
cae Bie) es raaea leer (6) 
Y cm ody * iy ete idly ; 
diac? da? 


By substituting these values of y—b, x —a, for y, —b, x,—a, 
respectively, in Eq. 2, we find 


Kqs. 6 will determine the position of the centre; and Kq. 
1, the length of the radius of the osculatory circle to the 
given curve at any point. When the curve at the point of 


286 DIFFERENTIAL CALCULUS. 


osculation is concave to the axis of x, as is the case if y is 
2 

positive and sa negative, then, to make p positive, we must 

take the minus sign written before the second member of (7). 
The first of Eqs. 5 indicates that the centre of the circle is 

in the normal to the curve at the point of osculation; and from 

the second of these equations we conclude that y — 6b and 
2 

da? 


the circle is always on the concave side of the curve, since’ 


must have opposite signs, and hence that the centre of 


y —bis the difference between the ordinate of the point of 
contact and the ordinate of the centre of the osculatory circle. 

In general, the contact of an osculatory circle is of the 
second order, that is, of an even order; and consequently it 
crosses the curve at the point of contact, except at particular 
points where the contact is of an order higher than the second. 

The osculatory circle is often called circle of curvature; and 


its centre and radius, the centre and radius of curvature. 


172. As an application of the formulas of the preceding 
article, let it be required to find the radius of curvature of 
a conic section at any point of the curve. If the curve be 
referred to one of its axes, and to the tangent through its ver- 
tex, as co-ordinate axes, its equation 
will be Ba: 

y? = pe + qa’, 
which, by two differentiations, gives 
d x ee dy\* 
ae ae vat ey me 
In the last of these, substituting for 
dy its value taken from the first, we 


dx 


have, — 


RADIUS OF CURVATURE. 287 


Ne D + 2pqx + ¢° a? 


nee y? Se 
whence 
Pie SP i. 
di y>” 
and for p we have 
dy?\3 
(+8) 
[ae : p? 


The numerator of this value of p is the cube of the normal 
NN’; for from the triangle MNWN’ we have 


aes dy’ dy? 8 
NN a ae et us n= y(1+ 3) 


and a 


Therefore the radius of curvature at any point of a conic sec- 
tion is equal to the cube of the normal at that point divided 
by the square of the semi-parameter. - 

The value of p expressed in terms of the constants of the 
equation, and the abscissa of the point J, is 


(q+97)2" + 2p + ge +p") 
p* 
173. The equation of the asa line to a curve at the 


p= 


point (x, y) being 
ay 
Gs AC — &3); 


the expression for the length of the perpendicular p let fall 
from the origin of co-ordinates on the tangent is 


288 DIFFERENTIAL CALCULUS. 


whence, by differentiation and reduction, 
2 2 2 
a! 1+ (4) ee mC oem 
dp aa” dx dx dx? \ da 
coe dy\*) 3 
{+ (a) 5 
dy\ d*y 
(« TY ze) as Cas roe 
So hae nae OC. AG +7) (Art. 171). 


+e) Ss 


And, if vr be the distance from the origin to the point of tan- 
gency, 


dr dy 
pp 2 aie e — ° 
mnt a hai To. =o 


and, substituting this value of x +- y in the expression for 
a we have 

dp _1 dr. ; _ i 

da. a! pdt an tikae dp 


174. If «andy are both functions of a third variable, s, 
then 


dy d?ydx d*x dy 
dy ds @y ‘dai ds det ds 
dx dx’? du? dx\? (1); 
i a) 


and these values of a a put in the formula for p (Art. 


171), give 
day* fdy\*)4 
UG) +@)3 , 
Py de dx dy (2). 


ds? ds ds* ds 
Supposing s to be the arc of the curve estimated from a fixed 


RADIUS OF CURVATURE. 


289 


d dy\? 
point, we find, from the formula ~ = J 1+ @) of Art. 161, 


da 1 — , (dae? , (dy? (de? ae 
eo (sey wt? ae es ds 
+@) 
da 
de ae dys? 
 edea re (3), 
1 1 dydx da dy ; 
dy dz diz dy’ p  ds* ds ds® ds ' 
ds* ds ds? ds 


From (3), by differentiating, we get 
dx d?x  dyd?y _ : 
Meegytc us dat 
Squaring (4) and (5), and adding, we find 
1 2a 2 d?y 2 
= (ae) + Ge). 
2 2 


d 
Eliminating ae abs 


sa? gr turn, between (4) and (5), observing 
8 
that 


da\*?  /dy\?__ 
ta) Ge)? 


we also find 


d*y da 
1 ds? ds? 
p ~~ “da dy 
ds ds 


175. To find the expression for the radius of curvature in 


terms of the polar co-ordinates of the curve, we substitute 
in the value of p, Art. 171, the values of —%, 


oy , given in 
dx’ dx 
Art. 131, thus getting 
37 


290 DIFFERENTIAL CALCULUS. 


dr\? ) 3 
aaa) 
dr\? Giy? 
42 (5) ~ ee 


1 
and, when r =~, we have 
U 


Ys fee 


dr  ldu dr 2 /du’ 1 du. 
desi? do’ do? a, u® do’ 
and these values, substituted in the above value of p, give 


_\e+G) | 


a u? ut So) 
do” 


176. The chord of curvature at any point of a curve is the 


8 
2 


portion of a secant line through that point that is included be- 
tween the point and the arc of the circle of curvature at the 
same point. 

The chord of curvature that, produced if necessary, passes 
through the pole, is obtained by multiplying 2p by the cosine 
of the angle included between the radius vector and the 
normal to the curve at the point; but if 7 is the radius vec- 
tor, and p the perpendicular let fall from the pole on the 


tangent to the curve, - is the cosine of the angle included be- 


tween the radius vector and the normal. But the value of p 


=~ 


is readily found to be ee hence the chord of cur- 
ie) 

do 
vature through the pole is equal to 


du\2 
2 
2u? + 2 GS 


ats ie wei. ON se 7 a 
2p say 2p Tp iteas mEEy (Arts. 173, 175). 


RADIUS OF. CURVATURE. 291 
177. Denoting by a@ the angle which the tangent to a curve 
at any point makes with the axis of abscisse, we have 


tan. & = a oe tan 19; 
da 
therefore 

d°y d°y 

CGT i da da da? 
dg dy\? ds (VF 

ee) ee 
' dx 1 
since —-- = - 


: theref . Se 
ne a ay 1erefore p= - 
(ae) 5 
178. The co-ordinates of the points of the curve at which 


the radius of curvature is a maximum or a minimum must be 


found from the equation of the curve and the equation — —0 
the latter leading to 


dy > dy d*y dy\? oe 
(5) diss dz} i+ (Gt) t= a 


Differentiating the second of Eqs. 3, Art. 171, we find 


dy, ay 
ie dic® + 

dy, a? dP y en a, 

3 - a aa 
dy, dx, da® x dic (2) 

da 
by Eqs. 4 of the same article. 

Comparing Eqs. dtgy _ aly 


= os Which 
da’ Bi 


proves, that, at the points of maximum or minimum cur- 


292 DIFFERENTIAL CALCULUS. 


vature, the osculatory circle has, with the given curve, a con- 
tact of the third order. | 

179. If a perpendicular be let 
fall from the origin of co-ordinates 
on the tangents drawn to the differ- 
ent points of any curve, as SIS’, the 
locus of the intersections of the per- 
pendiculars with the tangents will 
be a new curve, the properties of 


which will depend on those of the 
given curve. 

Denote the co-ordinates of the new curve by 2, y,; then 
will the length of the perpendicular p,, from the origin to the 
tangent drawn to this curve at the point corresponding to the 
point (x, 7) of the given curve, have for its expression 


1 


Pi ole ee 
Y1 
i — 
J te Ge 
The equation of the tangent to the given curve is 
d 
mn ea (vy — 2), 
wand » being the general co-ordinates. Since the point (a, ¥;) 
is on this tangent, 


di 
Yi — y= (a, — %). 


The equation of Op is u = hy ; and, because Op is perpendic- 
1 
da | 
ular to Mp, = = — ay whence 


(Yi—y)y¥i = —2(x,— 2), 
or YY + ee, =a + y?. 


EVOLUTES OF PLANE CURVES. 293 
Differentiating this last with respect to x, we find 


d d = 
He by os i ay to = Dy, <a) + 2 2, —. 


Substituting for ee its value, — re , transposing and redu- 
w 1 


cing, we have 


and, by means of this, »,; becomes 


Win ie 
r being the distance from the origin to the point (a, y) of the 
given curve, and p the perpendicular Op let fall from the ori- 


gin on the tangent to the curve at the same point. 


180. If f(x, y) = 9 be the equation of a curve, it has been 
shown (Art. 171), that, calling yu, », the co-ordinates of the cen- 
tre of curvature corresponding to the point (2, y) of the given 


curve, we have 


d 
w— p+ (y—9) 5% =90 (1), 


1+ (2 s) +(y—")54=0 (2). 


By means of these equations, the equation of the curve, and its 
first and second differential equations, we may eliminate 2, y, 


4 : ‘i and find a direct relation between » and». This will 
be the equation of a new curve, called, with reference to the 
given curve, the evolute; the given curve being the involute. 


It is evident that » and » may be considered as functions of 


294 DIFFERENTIAL CALCULUS. 


a; and, if Eq. 1 be differentiated under this supposition, we 


, 
dy\? d?y du dy dy __ 
ce?) ae da da 


and, through oer this reduces to 


have 


dy dy __ . dy diy es 
iB ae a mete i) 1 


whence, by the substitution of the value of dy derived from 


da 
the last of these equations, Eq. 1 becomes 


d 
yoo =, Cae 


These relations show that the tangent to the evolute is a nor- 
mal to the corresponding point of the involute, and the con- 
verse. 

A consequence of this property is, that the evolute of a 
curve is the locus of the 
intersections of the con- 
secutive normals to this 
curve. Tor take the two 
normals MK, MK’, which, 
by what precedes, are tan- 
gent to the evolute at the 
points K, K’. When the 
point J/’ is made to ap- 
proach the point M, the line M’K’ approaches the line MK, 
and the points A’ and N tend to unite in the point K: hence 


the point K may be regarded as the intersection of the normal 

MK with the normal indefinitely near or consecutive to it. 
Another important consequence is, that the length of the 

arc of the evolute between two centres of curvature is the 


EVOLUTES AND INVOLUTES. 2.95 


difference of the corresponding radii of curvature. To prove 


this, differentiate the equation 
pt = (en + (y— 9) 
treating y, “, 7, and p as functions of x: we thus have 
d, du d. di: 
p= @-M(1-F)+0-9 (4-2), 
x dx 
which, by Eq. 1, reduces to 


pe — HS —w—) & (a). 


. du dy dy it 
From Kg. 1 and the equation as + ap ts 0, we get 


ae du\? dy\? ie 
aes Sa 1d. 
Ea (=) +(Z) ae (b) 
— —— YP TG - 
_ iad | lento yf 
when s denotes the length of the arc of the evolute estimated 
from any point 2” (Cor. 2, Art. 161): whence 


dl dy 
Be te Oc hae 1 ae 


Co w= pa © 
MET ee smbination of Hos: aando, we find 2 = 4-2n 
z da da 
ee (8 = 7) : 
wherefore, since a 0, it follows that s = p is equal to 


some constant which we will denote by J; that is, 
ee p |, 6’ p!—1; .*,.8— 8’ =p —p’, 

or arc MK’ — arc PK = MK — MK’ =arc KK’. 

Suppose a flexible but inextensible string, of a length equal 
to M’K’ plus the arc K’KF,, fastened by one of its ends to F, 
to envelope the curve FAK’, and then pass in the direction of 
the tangent to the curve at K’ from K’ to I’. If this string 
be unwound from this curve, its free end will describe the 


296 DIFFERENTIAL CALCULUS. 


curve MW’S. Itis from this property that the terms “evolute” 
and “involute”’ are derived. Itis also seen that there may be 
an unlimited number of involutes answering to the same evo- 
lute KK’, and that, to describe them, it is only necessary to 
lengthen or shorten arbitrarily the part of the string that ex- 
tends in the tangent to the evolute. Since the tangents to 
the evolute are normals to all these involutes, it follows that. 
the latter curves have the same normals and the same centres 
of curvature, and that the parts of the common normals in- 
cluded between any two will be equal: hence one involute 


enables us to find all the others. 


181. Radius of curvature and evolute of the ellipse. 

The equation of an ellipse, referred to its centre and axes, is 
a2y? + b?x? = a?b?: 

dy ..\ Ba, Oy i eee 

da ay’ da® ay 

These values, substituted in the formula for the radius of 


whence 


curvature, Art. 171, give 


b4 
Mig 5 (bta? + aty?)3 
b4 atot 


aye | 
To find the equation of the evolute of the ellipse, resume 


the equations 


al 
x—p+(y—) =~ =0 (1), 


ady\? ie 
1+ (2) +u—)g4=0 @) 


fy ; 4 6. their values, it be- 


aty® + bta?y — a?bt(y—v) =0: 


of Art. 180. Putting in Hq. 2, for 


comes 


EVOLUTE OF THE ELLIPSE. 297 


ae yey fond Pe Fe ree Oat 
Re eres bia) y aly! bia eyes 


abt abt 
_(@ aby? $5 
stn hae te 
Making a* — b*? = c?, we find 
b4 ot) 2 C4 3 Cc? 3 
y—v=! TE a og pa ‘ y=! (3). 


Substituting in (1) the value of y —» just found, we get, after 
reduction, and the elimination of y by means of the equation 
ols 


of the ellipse, p= (4). 


Kq. 4 might have been derived from (8) by changing the sign 
of the latter, and in it writing a for y, anda for b. This isa 
consequence of the symmetry of the equation of the ellipse, 
and the relation between a, 0b, and ce. 


2 2 k ve 
Put aM Gat thon 2 = (4), $= (2) 
a a m 


Ch 


: x ‘ 
this becomes, when the above values of —, 3 are substituted, 
a 


3 

bn) + ( 
for the equation of the evo- 
Jute. The form of this equa- 
tion shows that the curve is 
symmetrical with respect to 
the axes of the ellipse. For 
y = 0 we have | 


The curve has, therefore, two 


-points, H, H’, in the transverse 
38 


298 DIFFERENTIAL CALCULUS. 


axis, situated between the foci, and equidistant from the cen- 


; 7 
tre. Making » = 0, we find »=>+n—=+ _ for the distances 


from the centre to the points H, E’, at which the curve meets 
the conjugate axis. 
By two differentiations, we find 


1 /u\-3 1 /\-t dy 
mnt ol Geen 


1 /u\-$ 1 /»\-4 /dy\? 1 /v\-3d*y 
a SY ey Moree tac — ee meets bs 
m° (“| n* () (=) ee n ) du? 


whence 


Since the numerator of this expression is positive, the sign of 


ad*y . ve tisy 
ae will be the same as that of the denominator; that is, du 
and » will have the same sign. The evolute at all its points 
will therefore be convex towards the axis of x (Art. 155). 


Moreover, we have 


I~ [ET 
nN nr 


Since this differential co-efficient becomes zero for » = 0, and. 


w\—-} J 
dy iG m m (= n 


infinite for « = 0, we conclude that the axes of the ellipse are 
tangents to the evolute at the points H, H’, and FE, E’; and 
that, in consequence of the symmetry of the curve with 
respect to the axes, these points are cusps. 
18%. Radius of curvature, and evolute of the parabola. 
When referred to the principal vertex as the origin of co- 


CURVATURE AND EVOLUTE OF PARABOLA. 299 


ordinates, the equation of the parabola is y¥? = 2pa, from which 
we find 

dy peatdige =p". 

<p = y Hee ane ye 
and, by means of these, the general value of p, Art. 171, be- 
comes, without respect to sign, 


2 


pt 
1+ 3 
( +5) OE Be 


= P i 
ae 
To get the equation of the evolute, we must substitute the 
d eae: 
values of - ; 7H ty 


d 
Bet (y—») 32 =0, 


dys” avy 
V+ (Se) ie eid peer ae 
which thus become 


opt (y—9) 7 =0, 


p* ig 
1 ge etre aa = 
The elimination of x and y between these equations and the 


equation of the parabola leads to the equation of the evolute. 


From the second, we find 


2 1” yey 3 
La git 05 Oe 
and, putting this value of v in the first, we have 
ye 
5) Basen eae OU; eae u— p= 32. 


Therefore we have 


300 : DIFFERENTIAL: CALCULUS. 


J ea 
y= =p, a 3 (H SD), Y* = pee 
») 3 
whence y® = p'r?, y° = (2px)? = PP) 


and prs oP? (hap )s 20 stp (u —p)” 
for the required equation. 

If the origin of co-ordinates be transferred to a point at 
the distance in the direction of positive abscissx, the new 
being parallel to the primitive axes, the equation of the evo- 


lute takes the form 


Par Oreo + Je 
We readily recognize that this curve is symmetrical with re- 
spect to the axis of abscisse, and that it extends without limit 
in the direction of # positive. | 


By differentiation, we find 


Fee be 3 | 

— = — 4, — =-— .. /— = ——' 

du 2N2%p"? due 4N2Ip"  ~ \/6p a/p 
Therefore, at the origin of co-ordinates, the axis of is tangent 


to the curve, and this point is a cusp; and, since the sign of 


2 
_ is the same as that of », the curve is at all points convex 
towards the axis of a. 

183. The expression for 


the radius of curvature and 


the equation of the evolute 


of the hyperbola may be de- 


| R’/ 0 x duced from those for the el- 
Lu u lipse by changing 6? into 
—b’, Thus we have, for the 


radius of curvature, 


CURVATURE AND EVOLUTE OF CYCLOID. 301] 


p= Oia? + aty?)! 
ity ath! ? 


and, for the equation of the evolute, 


The form of this equation shows that the evolute of the hy- 
perbola is composed of two branches of unlimited extent, and 
symmetrical with respect to both axes of the hyperbola. It 
has two cusps situated on the transverse axis beyond the foci, 
and is convex at all points towards the transverse axis. 

184, Radius of curvature and evolute of the cycloid. 

d 
2 , which, for this curve, is 
d or — 
-" ze a mii d (Art. 146), and differentiating, we find 
0 y : 


By squaring the value of 


Gy diy ar.dy. pe an r 
PPS eis 


Sean eye det dete 


ore ida. 
Substituting these values of dx? der? the general expres- 


2 


sion for p, we have 


a Y aaeAk, ‘ie? 3 sop = AA Ory. 


Ria nara 
y" 

Now, Pm = IN; and, 

from the right-angled 


triangle PNG, we 
have 


PN=VGNx NI; 
that is, PN=/ 2ry. 
Hence the radius 


302 DIFFERENTIAL CALCULUS. 


of curvature at any point of the cycloid is twice the normal 
at that point; and, if PN be produced until VQ = PW, the 
point @ will be the centre of curvature. 

185. The property just demonstrated leads, by very sim- 
ple deductions, to the determination of the evolute of the — 
cycloid. 

Produce the vertical diameter GN of the generating circle 
(figure last article), making VZ = GW, and on NZ, as a diame- 
ter, describe a circle. Through Z, the lower extremity of this 
diameter, draw LL parallel to Ox, meeting the axis O’D, pro- 
duced in &. The arcs PN, NQ, belonging to equal chords, 
gre equal; .*. arc VQ = ON: but OD = are Oe 
LQ=ND=LE. Thus it is seen, that if two equal cir- 
cles lying in the same plane be tangent to each other, and the 
one be rolled on the common tangent while the other is rolled 
on a parallel to it at the distance.of the diameter of the circle, 
the points of the two circumferences which are common at the 
time of starting will, during the motion, generate two equal 
cycloids; that generated by the point in the circumference of 
the second circle being the evolute of that generated by the . 
point in the first. 

This relation between the two cycloids, generated as just 
described, may also be inferred from the property of the sup- 
plementary chords of the generating circle, which are drawn 
through the extremities of the vertical diameter of this circle 
in any of its positions, and the corresponding point of the cy-_ 
cloid (Art. 146). For, since PG is tangent to the cycloid 
0O0’B at the point P, NQ, or PN produced, is tangent to the 
cycloid OQH at the point Q. Hence this last curve is the 
locus of the intersections of the consecutive normals to the cy- 


cloid OO’B, and is therefore its evolute. 


EVOLUTE OF THE CYCLOID. 303 


186. The application of the formulas of Art. 180 leads to 
the same result. 


From the equation of the cycloid, we have 


MS eset) Pe Te 


da TROL ke 
dy d’y . 
Substituting these values of a ek , in the equations 
tiie Chine 


d 
o—nt(y—) = 0, 


dy\? a? 
1+ (P)+0— = 0, 


we find from the second 


2 
Naar ts Bie sean raat ia 


vate y=—v (1); 
and from the first 


ee 
e—et(y—r) | —¥ =o. 


which, if we replace y by the value just found for it, and trans- 
_ pose, becomes 


The equation of the cycloid 
Shae) 


td 


2—rcos.7! 


—A/2ry—y? (3), 


by the substitution of these values of w and y, becomes 


2 gL fin 
pu +2» |— OS eee dene ays —2rv—v? (4), 

y r 

which is the equation of the evolute. But from Eqs. 1 and 2, 
it is seen, as it also is from (4), that there are no points of the 


804 DIFFERENTIAL CALCULUS. 


curve for which » is positive. Making » negative, transposing 
and reducing, we have, finally, 


B= Cone 


(5). 


Now, let the reference of the 
curve be changed from Oa, Oy, 
to Ha’, Ey’; positive abscissa 
being estimated from JL’ to- 
wards a’, and positive ordi- 
nates from # towards y’, 2 
and y, denoting the new co-or- 
dinates of the evolute. Since 
OD=a2r, DE = 2r, we have 
— OD — DI=ar —%,, y= JIF— FQ=2r—-y. 


Eq. 5, by the substitution of these values of u and », becomes 


Or we EE eee ee 
uv —Xx,=Pr ct ig oe te V/ 2x (27 — y1) — (2r—y)’, 


which reduces to 


ara, = reo = Y LAW ry, — y® Pee 
or ny =r (a— cos 2) — VB =H 
r 
But cos.~} a —= 2 — cos.7} a, introducing this in the 


equation above, it becomes, finally, 


Tt — = ib a eee 


This equation differs in no respect from (3), except in having 
21, Y;, instead of « and y; which shows that the evolute of a 
cycloid is an equal cycloid, situated, with reference to the axes 
Ex’, Py’, as the involute is with respect to the axes Ox, Oy. 
18%. It has been proved (Art. 180) that the length of an 


ENVELOPES OF PLANE CURVES. 8305 


* 


are of the evolute to any curve is the difference of the radii 
of curvature corresponding to the extreme points of the arc. 
In the cycloid at the point O (last figure), p =2V 2ry = 0: 
hence PQ = 2PN is the length of the arc OQ; and, with re- 
spect to the given cycloid, are PO’ = 2PG. 

To express the arc PO’ in terms of the ordinate of the point 
P, we have 


arc PO’ = 2PG = 2 2r x GO. 
But GC=2r—y: .*. arc PO! = 20/4r? — 2ry. 


Making y = 0, in this value of PO’, we have arc O0/O = 4r: 
hence the entire arc of the cycloid ts four times the diameter of 


the generating circle. 


188. Envelopes. If one or more of the constants enter- 
ing the equation of a curve be changed in value, we shall have 
a new curve, differing in position and dimensions from the 
given curve, but agreeing with it in kind: that is, if the given 
curve be an ellipse, the new curve will be an ellipse; if a pa- 
rabola, the new curve will be a parabola. The constants which 
thus change in value are called the variable parameters of the 
curve represented by the equation. 

The locus of the intersections, if any, of the consecutive 
curves of the same species, — that is, of curves whose equa- 
tions are derived from a given equation by causing one or 
more of its constants to vary by continuous degrees, — is 
called an envelope. 

Suppose F(x, y,a)=0 (1) to be the equation of a curve 
involving, among others, the constant a; and let a be taken as 
the variable parameter. Changing a into a-+h, the equation 
becomes F(x, y,a-+h)=0 (2), which represents another 
curve belonging to the family of that represented by (1). 

39 


306 DIFFERENTIAL CALCULUS. 


By Art. 56, Eq. 2 may be put under the form | — 
E(x, y, a) + h¥"’ (ax, y, a + 6h) = 0+. (3). 
Observing that /” signifies the derivative, with respect to a, 


of the function symbolized by F’, Eqs. 1 and 3, when simulta- 
neous, are equivalent to 

Bia, ya) = 0, 2" (a, 9,0 + oh) 
and the values of « and y, determined by the combination of 
these equations, will be the co-ordinates of the intersection 
of the curves of which (1) and (3) are the equations. 

If h be diminished without limit, Eqs. 4 become 

H(a, y, a) = 0, F’ (a, y, a)=0 (5); 

and the point determined by these equations is the limit of 
the intersections of the curves of which (1) and (2) are the 
equations. The equation which results from the elimination 
of a between Eqs. 5 will evidently be the envelope of the 
family of curves represented by the equation F(a, y, a) = 0, 
and of which the individual curves are formed ‘by assigning 
different values to a. 

The envelope touches each curve of the series at the point 
common to the curve and the envelope. This is proved by 
showing that the envelope and the curve, at the common 
point, have the same tangent. 

Since (1) becomes the equation of the envelope when in it 
the value of a, deduced from the second of Eqs. 5, is substi- 
tuted, let (1) be differentiated under this supposition, treating 
« as the independent variable, and a as a function of # and y, 


and we have for finding the value of ey for the envelope, 
dF dy “la Ts da aA 


dF | 
soa al Ly ely ttc Miata 6). 
pa aide obi ie aidy da): 0 yee 


ENVELOPES OF PLANE CURVES. 307 


But, at the point of intersection of the envelope with the given 


curve 
dF ’ nee 
da =F" (x, y, a) =0; 
hence (6) reduces to 
dk dF dy 


which is the same as that obtained by the differentiation of 
(1): whence, at the common point, the tangent line to the en- 
velope is also a tangent line to the given curve. | 

Ex. 1. Find the envelope of the family of straight lines 


derived from the equation y= aa + by causing a to vary. 


Differentiating with respect to a, a and y being constant, 
we have 
z——=0; ae a= | 
y= b2Vmea, y?=4me: 
hence the envelope is a parabola. 
Ex. 2. Find the envelope of the straight lines represented » 
by the equation y= aa -+ (b?a*+-c’)?, when a is made to 


vary. 
Differentiating with respect to a, we find 
2 
De ss a fe oe 
(b?a? + ¢?)2 b (b? — a): a?) 3 


Substituting this value of a in the given equation, we have, 


after reduction, 


which is the equation of an ellipse referred to its centre and 
axes. ? 

In each of the examples just given, it has been required to 
determine the curve from the general equation of the tangent 


308 DIFFERENTIAL CALCULUS. 


line. This process, being the inverse of that for finding the 
equation of the tangent line, is sometimes called “the inverse 
method of tangents.” 

If a point be taken on the axis of x at a distance from the 


origin equal to m, and a line be drawn through this point, 


making, with the axis of x, an angle having Ry for its tan- 
a : 
A ie 1 me 
gent, the equation of this line is y = — — (aw— m), and it inter- 
a 


sects the axis of y at the distance - from the origin. The 
equation of the perpendicular to this line, at its point of. inter- 
section with the axis of y, is y= ax+ ot Hence the geo- 


metrical interpretation of Ex. 1 is, “ From a point in the axis 
of x, at the distance m from the origin, draw lines intersect- 
ing the axis of y, and to these, at their points of intersection 
with the axis of y, draw perpendiculars; required the enve- 
lope of these perpendiculars:” and that of Ex. 2, “To find the 


envelope of a series of straight ines, so drawn that the product 


of the two ordinates of any one of these lines corresponding — 


to the abscisse,, -+ b, — b, shall be equal to c?.” 
Ex. 8. Find the envelope of all the parabolas given by the 


2 
equation ¥ = ax — Ie xv, by causing a to vary. 
Differentiating with respect to a, we have 
; 2 ; 
0=ae—— Pee aa? 
2 


whence, by substituting this value of a in the given equation, 


we find, for the envelope, 
at = (FE —y), or 2” + 2py — p= 0, 


which is the equation of a parabola. 


at 


ENVELOPES OF PLANE CURVES. 809 


Bx, 4. Find the envelope of the normal draw to the dif- 
ferent points of a given curve. 4 | 
Let the equation of the curve be y = =Se) then the equa. 


N 


tion of the normal is 
m—e@+(yi—y) s%=0 (1) 


in which 2,, y,, are the running co-ordinates of the normal 


From the equation y =/(a), y and oy can be expressed in 


terms of x, and thus x becomes the variable parameter in 


. t ff 


< 


~ . 


Kq. 1. Hence the equation of the required envelope may be 
found by eliminating x between (1), and 
avy (dy\" : 
ee ay) — ae (+); 
which we get by differentiating (1) with respect to a. 

Comparing (1) and (2) with the formulas, Art. 171, it is 
seen that x,, y,, are the co-ordinates of the centre of curvature 
of the point (xz, y) of the given curve; that is, the envelope 
of the normals of a curve vs the evolute of the curve. 

189. When the equation, representing the family of curves 
whose envelope is sought, involves several, say variable pa- 
rameters, and these parameters are connected by n —1 inde- 
pendent equations, instead of effecting the elimination of n—1 
parameters, and then differentiating with respect to that which 
remains, we may proceed as follows: Let the equation of the 
curve be 

Mie, Ue Cee yien 0, (ly. 


and let the »—1 equations of condition for the parameter be 


fila, 0 Sl Gast: 
ae b Crs me hea (2). 


Pan by b, Ges = —s 0 


310 DIFFERENTIAL CALCULUS. 


By reason of Eqs. 2, n — 1 of the parameters may be regarded 
as functions of the remaining one taken as independent. Let 
this be a, and differentiate Eqs. 1 and 2 with respect to it, 
thus getting 

Gf earidd ok ae 


da ' db da. dedat = 
df, df db df, de ] 


Hae wena da de da 


df, df, db PRO Sh sr 3 
da.‘ db. da’, de da 
fio a fines Opie 0) ene ia 

dai db da “de-da i on 
Now, it is plain that if, in Eqs. 1 and 3, all the variable para- 


meters and their functions be expressed in terms of a, and a 
be then eliminated between these two equations, the resulting 
equation will be that of the envelope. To effect this elimina- 
tion, we have 2n equations ; viz., the nm given equations, and 
their n differential equations: but there are only 2n — 1 quan- 


tities to eliminate; viz., the n quantities a, b,c..., and the 


ab de So: 
nm — 1 quantities -—— .: hence the elimination is possible. 


da’ da 
Multiply the first of Eqs. 4 by the indeterminate 2,, the sec- 


ond by 2,, and so on, and add the results and Eq. 3 together: 


we thus get 


dF df, df, Of ee ae 
ed AIS ly WAC URE pes Gay Go fede 
da M da es da ieK da 
ieee dh Uf, db 
Adie aur re a ai. 
+ (Gta gt = + tae) da | 0 6 
df dfs df, 1 de 
Ered : ace 
2 fz “a ‘de Te de AI iad os de , da 


ENVELOPES OF PLANE CURVES. Bi ba | 


By méans of the n— 1 indeterminate multipliers 4,, 2... .,4, 1, 


we may satisfy n—1 conditions. Let these be that the co-ef- 


ficients of a a ..., in Eq. 5, shall reduce to zero. These, 


da’ da 
together with that expressed by Kq. 5 itself, lead to 
pie p pce kets ‘ee x re 
da i lda aa 2 da an =a = n—1 da 0 
dk df; df, Un 1 
Os aly CS ai Ey Pela a ae d + = () 
Pee a Sao egg (6). 
dF df, df, OF gi 
La apie y ig Parkes ri ———— 
de “ 1 de 7 2 de n. “4 n—1 de 0 
J 
We have now the 2n —1 quantities a, b, c..., 41, Agee + An—ty 


to eliminate between the 2n equations (1), (2), and (6); and 
the result, being an equation between z and y only, will be the 
equation of the required envelope. 

190. When the general equation of the family of curves 
contains only two variable parameters, and they are connected 
by one equation, the process admits of the simplification, and 
the result takes a form the same as those in Art. 128. 

Ex. 1. Find the envelope to the different positions of a 
straight line of a given length extending from the axis of « to 
the axis of ¥. 

Let c be the length of the line, and a and b be the inter- 
cepts on the axes of x and y respectively; then the equation 
of the line is 


ET ae 
ee na) 
and the equation connecting a and 6 is 


a” + Ny Aped g (2). 


DIFFERENTIAL CALCULUS. - 


312 
Differentiating (1) and (2) with respect to a and b, a being 
taken as independent, we have 
a  y db db 7 
—+ 5 —= b— = 
a! b da Woes! da 
and therefore, according to Art. 128, 
oC y x y 
a? b? CB 1 
@ub a or ae 


b 
shyt aoa 
whence a=«atc3, b= y%c3, and 


@4+b=c=(rit+yi)ct: cb ty 
is the equation of the envelope. The 
figure represents the curve traced in 
the several angles of the co-ordinate 
axes. 

Ex. 2. Find the envelope of the se- 
ries of ellipses formed by varying a 
and 6 in the equation | 

0? On 
a) eae 


a and b being subject to the condition ab = c?. 
tiating with respect to a and b, regarding a as independent, 


By differen- 


we have 
pers edb Lele 
— ~-—=—0, = - —- == Qs 
aa nats We Rach j 
90? 2 1 
ee 3 ee, bia 
whence 


OU @? 
which is the equation of an hyperbola referred to its centre, 


and asymptotes as axes. 


EXAMPLES. Bl is 


EXAMPLES. 


1. What is the radius of curvature of the curve 
y = «vt — 4e* — 182? 

at the origin of co-ordinates? 1 
ATs p= ae 


2. Find the parabola which has the most intimate contact 
x3 
with the curve y = “ at the point having a for its abscissa, 


the axis of the parabola being parallel to the axis of y. 


2 
Ans. ( _- 7 a aly - i) 


3. Show that, at one of the points where y=0 in the curve 
9. ax (x — 3a) 
- we—4a 


] 


U) 
: “ OG od 
the radius of curvature is - = and at the other, 5" 


4, What is the radius of curvature of the spiral of Archime- 
des, the polar equation of this spiral being r = a6? 

Rn Ceara 

Qa? 1 72 


5. The Lemniscata of Bernoulli is the locus of the points in 


Ans. pr 


which the tangents at the different points of an equilateral 
hyperbola are intersected by the perpendiculars let fall upon 
them from the centre of the hyperbola. Its polar equation is 
r*?=—a’cos.26. What are the radius of curvature and the 


chord of curvature at any point of this curve? 


1 


Qo! bo 


2 
a 
ANS. ps an} chord of curvature = 
oO 


se eee 3 
6. Ifa curve have y = - G +e :) for its equation, prove 
oD 


that the general co-ordinates of its centre of curvature are 


ay? 
n=e—y |G 1, yi=2y. 


40 


314 DIFFERENTIAL CALCULUS. 


T. What is the envelope of all ellipses having a constant 
area, the axes being coincident? 

Ans. 4x?y? = c*; mc? being the given area. 

8. Find the envelope of the curves represented by the equa- 


weucw 


a and b being the variable parameters connected by the equa- 
b 2 
+@)=" 
wo y” 
Ans. 7p a ae A, 


9. Find the envelope of the system of straight lines con- 


tion 


tion 


necting, pair by pair, the feet of the perpendiculars let fall 
from the different points of an ellipse upon its axes; the equa- 
-tion of the ellipse being 


27 


© 


a" 


bw 


ie 
+5 =1. 


Ans. ey a (7) er 1. 


10. What is the envelope of the series of circles, the: circum- 
ferences of which pass through the origin, and which have 
their centres on the curve of which the equation is 

a®y” — b? (2ax — x?) = 0? 
Ans. (x? + y? — 2ax)? — 4a?x* — 46°74? = 0, 


INTEGRAL CALCULUS. 


ee eS oe we lee 
SECTION I. 


MEANING OF INTEGRATION. — NOTATION. 


DEFINITE AND INDEFI- 
NITE INTEGRALS. — DIRECT INTEGRATION OF EXPLICIT FUNC- 
TIONS OF A SINGLE VARIABLE. — INTEGRATION. OF A SUM. — 
INTEGRATION BY PARTS. — BY SUBSTITUTION. 


191. Any given function of a single variable may always 
be regarded as the differential co-efficient of some other func- 
tion of the same variable; that is, there is some second func- 
tion, which, when differentiated, will have the given function 
for its differential co-efficient. 

For let f(x) be the given function. If this admits of possi- 
ble values for real values of a, 
we may construct the curve 
CPD, which, referred to the 
rectangular axes Ox, Oy, has 
y =f (a) for its equation. The 
area included between this 


curve and the axis of x, that is ; 

limited on the one side by the fixed ordinate CA, correspond- 
ing to «=a, and on the other by the ordinate PJM, corre- 
sponding to the variable abscissa a, is evidently a function of 


315 


316 INTEGRAL CALCULUS. 


x; and, of this function, y or f(x) is the differential co-efficient 
(Art. 164): hence we should have 


d 
a (area dC PM) = f(a), 


.. (area ACPI) dx = f(x) dz. 


192. It will be found that the operations of the Integral | 


Calculus are mainly those of passing from given functions to 
others, which, by differentiation, would produce the given 
functions. The fact that these operations are the inverse of 
those of the Differential Calculus has been taken as the basis 
of the definition of the Integral Calculus. But the fundamen- 
tal proposition of the Integral Calculus is the summation of a 
certain infinite series of infinitely small terms. To effect this 
summation, we must generally know the function of which a 
given function is the differential co-efficient. The proposition 
may be stated thus : — 

Let f(x) be a function of x, which is finite and continuous 
for all values of x between a , x,, and of invariable sign be- 
tween these limits. Let x, be greater than ~,, and divide the 
difference x, —,) into a number v of parts, equal or unequal, 
represented by x,—%), X,—% , %3—%q..., ©, —@,_ 13 Te 
quired the sum of the series 

S =f (%o) (%1 — &o) +/ (#1) (22-21) + + 

+P F@,S3) (Ca 
when the number of parts into which x, —, 1s divided is in 
creased without limit, or n is made infinite. For brevity, 
by 


denote the intervals 2,—2a), @,—%,..., L,—2y_1, 


h,, hy..., h,, and the series becomes 


S=f(xo)hy +f (@i)he +e +P (Cn—a)hnitf(En-1) ha (1). 


7.) = 


MEANING OF INTEGRATION. 317 


Now suppose /(x) to be the function of w, of which f(2) is 
the first derived function ; then 


tion et h) — F(x) 


=f («). 
But, before passing to the ud we should have 


ae te =A) — p(w) 4 p (Art. 15), 


p being a quantity that vanishes with 1: therefore 
Fle +h)— F(z) =h{ f(z) +0} (2). 
In (2), giving to’ the values h,, h,..., h,, and to & the 


values &), @,..., ©,_1, &,, and denoting the corresponding 
values of p by pj, po---, Pn, observing that 
ty thy=a,, e%+h,=2..., 
we have 
F(x;) —F (a) =h, WACZ) +pxi}, 
F(a) — ate =h, ve ede 


P(e, Bere... ) aif Mee ee {> 
F(@,) — Fl@a—1) = ba {f(@n—1) + Po}. 
Adding these equations member to member, for the first 
member of the result, we have F'(x,) — F'(x)). The second 
member is composed of two series, the terms of one being of 
the form hf(x); and of the other, hp. Denote the sum of the 
terms of the first series by 3/(x)h, and of the second by Sph; 
then our result may be written 
F(2c,) — F(a) = 3f(a)h+3ph (3). 
If p’, the greatest among the quantities p,, p,...,p,, be sub- 
stituted for p in the series represented by Sph, we should have 


Sph <p/(hyt+hy,+--» +h,) =p! (x, — 2). 


318 INTEGRAL CALCULUS. 


But p’ vanishes when / is decreased without limit: hence 
I(%,) — #(29) 
as the value towards which the series Sf(x)h converges when 
the quantities of which h is the type are diminished without 
limit ; that is, 
lim. £f(x)h = F(x,) — F(x,) (4). 

193. lt may be readily proved that 2/(a)h has a definite 
value when / is indefinitely decreased, and when, therefore, 
the number of parts into which the interval a, —a, is divided 
becomes infinite. For let 4, be the least, and A; the greatest, 
of the values assumed by /(x) for values of x between a, &,,: 
then 

af(x)h >Ay (hy thot +++ + hy) = Ay (L_ — 2p), 
aAf(a)h<A, (hj th.+--- +h,) = A,(x, — Xo); 
and since, by hypothesis, both A, and A, are finite, the same 
is true of S/(x)h. It is evident that the values of /(x) inter- 
mediate to A), A,, will be furnished by the expression 


S\ ey) +6(a, —a)}, 
6 being a proper fraction; and that such a value can be as- 
signed to 6 as will make 


=f (x)h — (x, A Lo) f} Xo a" 0 (©, ay ay) | 
a true equation. 
194, Putting Eq. 3 of Art. 192 under the form 
Sf(a)h = F(a,) — F(a) — Sph, 
it is seen that the value of Sf(x)h will, in general, depend on 
the number and value of ‘the parts h,, h,...,h,, into which 


the interval x, — x, is divided, but that lim. 3/(x)h, for which 
<ph vanishes, is independent of the mode of division. When 


DEFINITE AND INDEFINITE INTEGRALS. 319 


all the parts into which x, — a, is divided are equal, each is 


Ln — Ly 


equal to ; and any one of the intermediate values of a, 


as @,, 18 equal to a, + ~ (x, — %,). In this case, the value of 

lim. £f(x)h is represented by J 7) dz = F'(x,) — F(x). 
a 

The symbol i. signifies swm, and dx represents the h = Aw of 


the expression +/(x)h. The quantity 
Jf (2) de = Fle.) — F(a). 


is called a definite integral; the operation by which we pass 
from f(x) dx to ii "f (2) dx is called integration; and x,, %o, 


are the limits of the integral. Since F'(x,) — F(x.) is the 
value of this definite integral, we must first find the function 
F(x) of x, of which f(x) is the differential co-efficient. The 
relation between f(x) and /’(x) is expressed by 


d 
which, by the notation of the Integral Calculus, is 


Jf (2) dx = F(x). 

195. The function f(x) of x, which, differentiated, would ° 
reproduce f(x) dx, is denominated indefinite integral. But 
a constant connected with a function by the sign plus or mi- 
‘nus disappears in differentiation; therefore the more general 
relation between /(x) and F(x) is 

J f(x) da = F(x) + C: 
so that the proper value of y to verify the equation 
, dy = f(x) da 

is given by the equation 


y = ff (a) da + 0; 


320 INTEGRAL CALCULUS. 


and the two symbols d and /, the one indicating differentia- 
tion, and the other integration, neutralize each other, and we 
shall always have 


fdu =u + C, df du = du. 


The constant thus added to an indefinite integral is called 
the arbitrary constant of integration, or, simply, the arbitrary 
constant ; it being any quantity which does not depend on the 
independent variable a. 

The operation of passing from an indefinite integral to a 
definite integral consists in substituting in the indefinite, suc- 
cessively; the limiting values of the independent variable, and 
taking the difference of the results. The arbitrary constant 


will, of course, disappear in the subtraction. 


196. In differentiation, constant factors may be written 
before the sign of differentiation. 'The same may be done in 
integration. For 


[dau = au, afdu = au; 
fdau = af du, or fadu=afdu; 
or, more generally, 


faf(x) da ne af f(a) dx. 


Observing that [Ao dex is the expression for the limit of 


the sum +/(x)Aw, that is, the expression for this sum, when 
the number of parts of the interval x, — x, is increased with- 
out limit, and the value of the parts severally correspondingly 
decreased, it is evident, that, at the limit, the addition or omis- 
sion of a finite number of the components /(x)aw of £f(x)ax 
would not affect the result. 


DIRECT INTEGRATION. 391 


A single term f(x) Ax of the expression 5/(x) Aa is called an 
element. 

197. Direct integration of simple functions. 

We shall, for the present, confine ourselves to the deter- 
mination of indefinite integrals, to which it must be understood 
that an arbitrary constant is to be added. 

There are many cases in which a function is at once recog- 
nized to be the differential co-efficient of another. In such 
cases, we have simply to write the second as the integral of 
the first. | 

Subjoined is a table of the integrals of the simple functions. 


geri az 
(ot a re 
n+] la 
fsin. ede == —c0s. 2, ferda moni off 
j dx 
fcos.adzx Sill, 2, — = lg, 
x 
i; dic ‘ dc pet iv 
eo = Lan. © ee SI i ee C08 S 
cos. x ; / a? — 2? rey a’ 


[aes ‘ {Reiida a PAR aot iS 
sina SS Ged O16 fF x, J/1 — a? SSS i Ole wv == — COs. a, 
dic 1 Lye 1 whe 
J ey gi = 0. ica nee = 
In all of these formulas, x may be the independent variable, 
or it may be any function of the independent variable ; for if, 


ay 
in the formula far da =n x be replaced by f(x), we 
should have 
n Di ACS ee 
ee lt thiula Pcsds ee reduces t 
1en n = —1, the ormula {x” ag a | reduces to 


ay. 


Saris 60 
ax 0 : 


41 


322 INTEGRAL CALCULUS. 


whereas, we know that J ee = lx. The failure of the formula 


to give the true result in this case arises from the fact that 
the transcendental quantity lx cannot be represented by an 
algebraic expression. It may, however, by a suitable trans- 
formation, be made to give the true value of f x" dae when 
ont 


per tin 


n=—1. Take the general formula fa"dx = 


which may be written 
‘do =P 
fe te : 


Now, the term ++ in the second member, may be in- 


re: n+ 
cluded in the arbitrary constant C: and i we have 
Jerde — 7 aa) Or 
or, omitting the constant, 
x Zs grtl PEE) xe 0 an 
fa (fr Sie ars Ce | when n = — 1. 


The true value of this is found by differentiating the nume- 
rator and denominator with respect to n, and taking the ratio 
of the differential co-efficients (Art. 101). We find 


i ie te Sf ntl 2) 
sss ee | Oar s lx. 
( n Ae 1 joel 1 /n=—l 


198. The rules of the Differential Calculus enable us to 
find the differential co-efficients of all known functions; but 
the inverse operation, of deducing the function of which a 
given function is the differential co-efficient, is not always pos- 
sible. Whatever the assumed function may be, there must be 
some other function of the quantities involved, which, differ- 
entiated, would produce it (Art. 191). The second of the two 
functions thus related as differential to integral may not be- 


INTEGRATION OF A SUM. 323. 


long to any of the small class of simple functions which have 

been admitted into analysis, or to any combination of such 

functions ; in which case, we are limited to series and approxi- 

mations for the expression of integrals. For example, we rec- 

ognize a to be the differential co-efficient of sin}, 
dx 


or that Ila sip. at because the latter function has 
a? —x 


been named, and its properties investigated. Had this not 


: da 
been done, the integral | 7; could not have been ex- 


pressed by means of a simple function. 
199. Integration of a sum of functions of the same vari- 
able. 
In the Differential Calculus (Art. 19), it is proved that if 
yY=S(e)EP(e) + Y(%)E..., 
| 
then “ =f’ (x) + 9/(@) +w'(x)+..., 
or dy = f'(x) dx + g'(x)dx+w’(x)dx...: 
whence 
fdy=y= ffi (x) dx + fo’ (x)da + fy'(a)da... : 
Hence the integral of the sum of any number of functions is 


the sum of the integrals of the component functions. For ex- 
ample, | 
Marr de OT 
Jf (4a + Ba ED OE ars nests Oy 
also 
f (5x4 — Tx*® + 4¢ —3)de= att at + 2x? — 3a, 
Si ee 

and ipa, 1 oe et — 2a? + dla. 

; v 


4 0 


324 INTEGRAL CALCULUS. 


200. Integration by parts. 
If w and v are functions of the same variable, we have, by 
differentiation, 


a(uv) du 
dx ue ate da’ 


The integration of both a eae of this gives 


w = fue Y da + foo! de: 


therefore 
dv du 
Jus de = uv — Jo da 
or fudv = uw — frdu. 
This method of integration, by which the determination of 


an integral Jude is reduced to that of another fvdu, is fre- 
quently employed, and is called integration by PHS 


Hix. A. fa? cos. «dx. 

Put 2? =u, cos.cdx = dv—=dsin.a; then, by the formula, 
fx? cos. ada = fad sin. = x sin. x — 2fasin. da, 
fasin.adx = — fadcos.x = —acos.a + f cos. xdx 


= — xcos.% + sin. &. 
We shall therefore have, by the substitution of this value in 
the first integral, 


fx cos. eda = x’ sin. w + 2x” cos. « — 2 sin. a. 
Ex. 2. fivda 
Make lx =u, dx =dv; then fladx = ala —a. 
a4 faure*da. 
Making x" = u, e*dx = dv, the formula gives 


fare de =e? n fat” Vena 


INTEGRATION BY SUBSTITUTION. 320 


and the integration of «"e*dx is thus brought to that of 
a”—le*dx. By another application of the formula, the expres- 
sion to be integrated would become a” ~*e*da; so that, if n be 
a positive whole number, the proposition would be reduced, 
after » applications of the formula, to finding the integral 
e*dx —de*. Hence, by a series of substitutions, we should 
have the required integral. 


Making n=l, fuer da = e*(x— 1), 
sf To a, feretdx = x e* — 2 fxe*da 
= e* (x? — 2x + 2). 

201. Integration by substitution. 

It is sometimes the case that a differential expression, 
Jf (x)dx, which is not immediately integrable, becomes so by 
replacing the independent variable by some function of a new 
variable. The function selected must be such that it shall be 
capable of assuming all the values of the variable for which it 
is substituted within the assigned limits of the integral. 


Let ¢ be the new variable, and suppose «= q(t); then, by 
the Differential Calculus, 


d 
a0 Oy or de = gi (t)dt, 
and f(a) dx =f {9 (é)}q! (t)at: 


whence, by integration, 


Sf (a) dx = ff ip} a’ (tat; 
in which it must be remembered, that, if the first integral is to 
be taken between the limits a and 8, the second is to be taken 


between the corresponding limits a’ and 0’. 


Ex. 1. [ (aa + b)’ da. 


- 826 INTEGRAL CALCULUS. 


Put ax +b=1t, whence da = “dt; and therefore 


leit; Sh eel u tiie} 
(ax +b) da = — ft dé = ae 


Replacing ¢ by its value, we have, for the required integral, 
1 (ax +b)" t! 


J (aa + by" des = — mn 
Ex. 2 wer 
Make 843 + 5 = 7, then 3x?dx = : dt, 
i oe 
therefore a <= ; (8a? + 5). 


It is evident from these two examples that success in effect- 
ing integration by substitution must depend on the ingenuity 
of the student, and his knowledge of the forms of the differ- 


entials of the simple functions. 


MISCELLANEOUS EXAMPLES. 


ada ae ed 
a = See r m4 2 —— “* a‘ e r 74 2 — 2 
ae a Make Va? + a =, 2. attat=t 


xdx —tdt; and therefore 


f xd ; 
Vat gl 

2. [Va — x dx. 

Putting Wa*— a? =u, «=v, and. integrating by parts 
(Art. 200), we have 


[Va — & eo? da =ar/q? — x? +f _ we 
a 


— o? 


(1). 


MISCELLANEOUS EXAMPLES. 327 


But [Vea de = [ae 
— 2 


a’ dx x da 
V/ a? — x” foe Va — 27 (). 
Therefore, by the addition of (1) and (2), 


of Va? — a da =an/a? — a+ a? de, 
V/ a — a?’ 


and since 
da 
/ a? — 2? — a? 


we have finally 


= a’ sin. mie AATUCLOT, 


[VG =e dx = 5 LA/ gq? — gw pay L gt 12, 
a 


a? 


d. pegs ae ene 
3. S Jae Make Va? + a? =t—a: .*. a? =t? — te. 


Hence, by differentiation, dx = : - “dt: therefore 
t— x dt 
lw 21g? =f > BS tex si oan 
= Ua + A/a? + a). 
4, l= Making Va? — a? =¢ ee) and proceed- 
ing as in Ex. 3, we should find 
da 
cm Faas Ot i: 


[Va? +a’ dx. Integrating by parts, Art. 200, we have 


[Ve +e da=aV/ax? + a? +2 — | yaaa ae (1). 


328 INTEGRAL CALCULUS. 


6 cee 2 2 
But ii 
x a 


aN ie 
Fe 2). 
Therefore, by the addition of (1) and (2), 


a pa dx 
of[Va? +a dia = an/ a a + a | 
By Ex. 3, 
dx 
a | ara = te Va 
and hence 


2 
SJ/2 + @ dz = oF EG +S Ya + opal). 
[V@ —a’ dex = Sa Fa @ — © ae + aa) 


The quantity under the radical sign 


Ua ay 


in the denominator may be put under the form 


otra Bort oak) 


2 2 
=(#+4) +q—%: 


hence 


+q-4 


Reece; SoUe ar ST ae 


Making x + £4, and q— Fat, we have dx = dt, and 


dt 
Vea 


In this last, substituting for ¢ and a their values, we have 


— ,= Ut + Vt? + a?) by Ex. 3. 


MISCELLANEOUS EXAMPLES. 329 


er Oe aa p> p 
eee tat Cts) +87) 


a1(0+h4+Va + pe +9) 


8. Beit ) aA me dsc. 
Put «+ P=, iti = a?, then 
Se pa fy de=JVP FO dt 
=SIWF EO +o t+ VE Pa), 
by Ex. 5. Replacing ¢ and a? by their values, we have, finally, 
Jv epee 9 do= 5 (2+5)Vi +p + 


1 : | 
=e 3( ae (e+$4+ve Fpe+4). 
ax 
9. ee. Letra —i;-thén Jar — 2’? =a? — ¢7, 
Iam — x? cabanas 
and dx = — dt: therefore 


Aga: 
<== -~(sa— ==; = COS. ne 


_;a—«x Sey 
Cs. = = Vors, 
a 


he, a 
10. lz Put Ges ee? then 
Lh eee! a" 
2ax — a? = fs Betty, waa — a) = eae) e asitad 
1 ay t 1—¢ 
dx =o and therefore 
adt 
f dx “ie (1 — Os ee 1 dt 


tr/ ax — a? — a(1+t) VAL, 2 
ia a ae 
42 


330 INTEGRAL CALCULUS. 


i Lie Y tere Lie 
= 1+ gin. ¢ — ~'sin, ~ See 
a a 


x 
fxcos.azde. Assume u=—, v=sin.aw; whence 
a 


dv =acos.axdx and [a cos.axdx = f udv: therefore 


x sin. ax sin. ax 
fa cos.axada = pind hea da 


Ped SIG “a COS. Aa 


a a? 


__ sin. AL 


12, fe sin. aada. Put u > ve; 


e es 


; sin. ax ae COS. Ax 
fe sin.axdax oa | dx. 
c c 
But we have, in like manner, 
ae" COS. ax poe an a? ay ax 
[US de= ee + fe emda: 
Cc 


hence 


. * 2. . 
sin. ax ACOS. AS ox ma: sin. ax e da, 


fe*sin.axda = eo . 
C | C c 


which, by transposition and reduction, becomes 


e°"(Csin.ax — a COS.ax) 


e** sin. axdxa = : 
i i re 


13. fe ‘cos, axda. 


Proceeding with this as with the last example, we should ~ 
find 


e°*(e cos.ax + asin. ax) 


fee cos.aade = * cia 
dix dx 
14. ih as SS a 
V (q+ pe — 2?) Jia 


2 
Put g-+4- =a’, wat: >, dt =o 


MISCELLANEOUS EXAMPLES. 331 
Therefore 


Rare ut hee 
ald rece ae 


P 

a pare ae 
ee: 2 Real eid S 
a 42 ar 

4 


15. fYatpe—ade=f |}q4+%—(2—$) tae 
2 
~ Making gto =a', % —f=1, we have 


IV (a + pe — x?) da Ree —?t? dt 


= sive 4 = in. mee by Ex. 2, 


1 
= — (22 — pw/4q 4+ p’ 4 E ihes the saa 
pee OV a Pr (4q-+2") vipers 
da 1 dt 
16. la ee whence da = — -,, and 
eas! 1 
a era ae renee a Aha 
Aa/ x =F fe a= av 1 ge 
f dx dt trae ae 
: xAa/ x? —-a3 V1 ~ Qt? eet 
a” 
—= —-sin. lat = — —sin.-!- 
But, since 
ORT pee asia ee 
sin ~ + cos. sin cos eGo: 


hence, throwing — 


may write 


INTEGRAL CALCULUS. — 


dx Nee dt 
17. loa Make w= 5: : . da = — =, and 
f dx PER: dt 


fT 1 
es Ss a a es ee a een 2 — 
pa J pee (t+ =5) 
d aetat 


a ax 
ee x 1 x 


| wn sae: a+ /a? + a? 
by including 54 in the constant of integration 
da 1 1 1 
18. ———. = — vee di. 
vee wal Gees a ks 


fey ah da = dix 


20. Ci — a 


c+ a 
1 eae 
st pg ere Pls 5 Meta) = - bed 
If x is less than a, then | 
f AGS dx ph dx da 
e—a?  4@—2  Y/\a—a%' ate 
14%, 
=> la—x)— 5a +2) = =: prite 
dic at+2z 
ang laa tet 
dix ur da 
2 ena 


(245) 49-4 


Suppose g — i to be negative, and make 


2 
e+f=tq—-t =a’; 


MISCELLANEOUS EXAMPLES. 90 


then, by the last example, 


ii dx a dt RB jbe, te 
miaepat+gq Y@—a? a tta 
i 20 +p — /4q — p? 


~ V4q— pp? 2a +p+V/4q—p* 


9 


“ 


If q = be positive, then g — apse a’; and 


das 1 et 
ie +pxe+q =/aeea= a a 


: -1_ 20 p 
V4q —p /4q — p” 
uP ne 
mx +n sora 2 2 
0 [ee en re 
a + px +- q a* + pa + g 


m 2a m dic 
es yaa 
2/7 «+ px+q 2 /¥ «w+ pe + q 


The integral of the first term is 5 U(x? + px + q), and that 
of the second is found by Ex. 19. 


21. i} es —f2 a als sare 


S1n. & 


Make cos.2 =¢; then 
dices res f 1h oes 1 1! 7) 1, 1+ cos.a 
cei e))606lCUl ULL ECC cone 
by Ex. 18: but 
1,1-+ cos.c a1 ea) = etm: 


> 2 1 —cos.a@ >. \1 + cos. a 


99. f da =f cos.cde f dsin.a 


Com ae cos; @ J 1 —sin’a2 


334 \TINTEGRAL OALCULOGS* 


C08. 5 aa aaa 


wd 1+ singe es 9 


2° 1—sin.a 1 ti | 
cos. 5% — sin. 5 & 


Seale at 
ne a7 +5)" 


da sin.2z + cos.2 a 
93. {———— =i A aE da. 
sin. & COS. x sin. 2 COS. x 


= f (tan. « + cot. a)da 


oo tan. 2. 


= —lcos.x+U sin. =e 


24. jouer =f eh oe a 


sin.” @ cos.? x sin.” x cos.” x 


= [ (sec? « + cosec.’ w)da 


— tan: « — cot: a 


dx 
a A) scares ee 


da 
a(sin2 5 + cos? 5) +b (cost — sin? 5) 


sec. 2 dex 


iio Se 
atb+t(a — b) tan? 
by observing that 
roe oC oie 
sin? 5 + cos. 5c 1; Cos. 2 = cone 5 i sin.? 5? 
and dividing the numerator and denominator of the result by 


cos. When a > 8, the last integral may be put under the 


form 


MISCELLANEOUS EXAMPLES. 


338 
x 
2 Bens ee ee x 
| —— tan.— ee LAT) 
ene 8s, fat aah lato ( ato ») 
a—b 2 a—b 


i} dx cee ee Creeigs {eee x 
atbcos.a/ys, VWa?—b? ” rey vee 5) 


When a < 6, we have 


x 
f , ts d tan. 5 
at+tbcos.« b—a b+a, 22 
b—a 
i Vb—atan.=+Vbba 
J a8 
vb o /b—¢ tan. — 5— vba 
by Ex. 18. 
ae 
a+b sin. a . ow 20 
a+ 26 sin. = cos. = 
sie 2 
ye dx 


a (sin? 5 + cos. 5) + 2b sin. 5 C085 


x 
sec? — 50% 


a(t +- tan. a ae: 5 


336 INTEGRAL CALCULUS. 


d (tan. 5 a ;) 


a 2, f2 2 
‘iu paleo +(tan.5 +5) 
Q: 2 ee 


2 Best a zc 6b 
my epieam ea Sa (tan. 5 eal 
when a > b; but, if a < b, then 


“% 2 ee 
f es teh 2 d (tan. +5) 


at+bsin a Ue ss 
( st) oy a 


Po a? 
tan.2 + 2 Vb — a 
a 
\_ Vb? — @ ton. 1 2 
was 
Le atan. 54+) —/b?—a? 
=a) OF ate 


mL |. 
Bian. | Oe Ne 


202. Rationalization and integration of irrational functions. 

Examples of integration by substitution have already been 
given: we now proceed to show under what conditions cer-. 
tain irrational differential expressions may, by proper substi- 
tutions, be rendered rational, and integrated by the methods 
previously investigated. 

Let us assume the form x(a + ban)a dx,in which m, n, p, 


and q are entire or fractional, positive or negative. 


© 
Put alam = st; a= (5 = 


gu?" 
1 
nbn 


I=n 
(27—a) ™ da: 


ai 


IRRATIONAL FUNCTIONS. 33% 


whence 
P i oe 
fara + bx")i da = ty f Pt 8-1(29 a Ye Eee 
nb * 
Ey. aaa oe 
Now, if oat is an integer, the binomial (2?—a)™ —' 


is rational in form, and may be expanded by the Binomial 
Formula into a finite number of terms when the exponent 


1 ; a : 
me —1 is positive. Each term of the expansion, being 
multiplied by z?+%~-!dz, will give rise to a series of monomial 
differentials which can be immediately integrated. 


We may also write 
fom (at ber) dex = fo" (bax-"ide; 
and, by comparing this with the first case, we conclude that 
the substitution of z4% for b + ax—” will reduce 
| Sam*a(d 4 ax-"); dic 


to an expression that is immediately integrable when 


m+tl p 
n qd 
; me: ; ar 7 m1 ip. ; 
1s a POBrEYS integer; 1.e., when ‘9 ae 1s a negative 


integer. | 
. 2M, L . 
Hence fom (a+ ban)e dx may be rationalized and _ inte- 


grated when = is a positive integer by substituting z? for 


we 


m+ 1 ‘ Fotis 
a+bx"; and, when __. +~ isa negative integer, by substi- 
q q 


tutingy 2? for b+ ax-". 
It will be shown in a subsequent section (2) that the inte- 


| tna ae 
grals may also be found when is a negative integer In 


. 43 


338 INTEGRAL CALCULUS. 


1 ree) 
the first case, and when aoe is a positive integer in 
n 


the second. 


2 ie 
fala + ba)’. Hereom=1, n=4, 7 oe 


1 
Beas = 2,a positive integer. 
n 
yl 
beavar a+ ba =2?: i ee r a day = 7, 


2% f x(a 4 bee)? dee =sfe — a)z*dz 


= 5 fe —aeh)de 
=7(--9) a+ bu)i ae 5): 


NTO b? 7 aa 
. é 
Ex. 2 f See ;: In this example, 
(a* +a)? 
m=8,n=2, 2 oe oe 2 
Yana AS ee 2\2 : ada, 
Pot a?-+ 2? = 2°: 0°. g= (2? — 9’) , and d7=er 
(2? — a)? 
x* da 
= | (2?—a*)dz == — az 
eS: J ) 3 
a. ee 
(at iets & 2a 
: 3 
Ex. 3 f——. In this case, m = 2, n=4 Boe 
(a? + 27)’ q 
pdt ae oe a negative integer. 


—. (1+ aa “Ede! 
a? + a)? 


IRRATIONAL FUNCTIONS. 339 


Let Si alee he a (ae eee 
Coal 
Tre azdz 
8 
Gal) 
GI er Ge a piek.o I 
: (a? + o:?)* aier tart 
os 
TE MEE tae te 
3a°(a? +0)” 
dx p 1 
Ex. 4. os Here m= —2, n=2,4+——- 
x1 +- 2)* q 2 
Bute) @=>— 2°: LS ina sees Ea dz—= — ad as 
(21) (x? — 1) 
d 
. | aes ta evar ake? kee cers 
“Ya@1ta) ~ «1+ —*)" 


Functions in which the only irrational parts are monomials 


can always be rationalized and integrated. Thus, suppose it 
is required to find 


1 2 
ns 2 ees x) dar 
8 : 

bee we" 
Putw—t*; .-. da — 6t°dt; and we have 


+a? —2£ el ere 
pe, 


ae —H+%+0—t+e? — 
a1 6 elt Mate +: ety, 
6 
Stee ED at Bae #h hs =! 
re +a +t ze + 2¢ 6¢ + 6 tan.— 1b, 


which becomes the integral in terms of x by replacing #by a, 


340 INTEGRAL CALCULUS. 


The rule to be observed for rationalizing such expressions is 
to substitute for the quantity under the radical sign a new 
variable affected with the least common multiple of the indices 
of the radicals for its exponent. 

Fractions in which the only radicals are the roots of the 
same binomial of the first degree may be reduced to the case 
just treated. 

For example, required 


xv* + (ax + b)} dex 


a + (ax +b)" 


1®—b 6t° dt 
Assume. aextb=?*: .°.%= - , dx = ——, 


a 
ak 
P hye a 


(axe + by? = t*;(ax-+-b)° = 
By these substitutions, the expression to be ec be- 
comes the rational fraction 

6 Ale by atti tde 

a? —b-+ at?’ 
The general method of integrating rational fractions will be 


investigated in the next section. 


EXAMPLES. 
: 3 + 2a 

—_ —1 

as = sin. ie | 
n+l 
IX fortede Se gi (to aoe ; 
n+ 1 n+] 

3. fo sin. 6d = sin. 6 — @ cos. 6. 
4, Ss stall... .6". 


sin. 2x 


Dom f (1 — cos. 2)'de = 5 —2sin.a 


MISCELLANEOUS EXAMPLES. 841 


a? dc Rea xs 
. Sma ws ara 
1+ cos. x : 
T. ee: Sin. 0% =U % + Bin. @). 
. e+ sin. # a 
8. 1+ cos.2 adr — ¢ tan. 5 
9, J—= AED Us 


10. fessin. ma cos. nada 


e“ asin.(m + Bie — (m-+ n)cos.(m + n)x 


id. a? + (m+n)? 
e* asin.(m — n)x — (m — n)cos.(m — n)x 
ee etna ew 


Having found the indefinite integral, the definite integral 
between assigned limits, except in special cases, can be at 


once determined. 


eh Cee oem 70,” 
atk Va? — «0? dx =? 
U 
xr/ a? — a” a? 
for ead Aires , +g sin = ¥(2); 
a? 
and w(a)=——» (0) = 0: +. yla) — ¥(0) == 
2a 12 3 us 
12. J ver. 4 ia eet 


By making « = a(1 — cos. 6), we find 
x 2 
fver- OY) bs Smee fas sin. ddd 
a 
— asin. 0d — a0 cos. 6. 


The limits 7 and 0 for the transformed integral correspond 
to the limits 2a and 0 for the given integral. 


342 INTEGRAL CALCULUS. 


2a 5a” 
ag = : 
13 i aver da 1 
da xe on 

: = -—_lIt a 
sin. @ + cos. x =F; st © a 3) 
i i} sin.” eda = (eto na Vatan. © & 

; a+ bcos.a ab? ia Vato b 


een atba? a $ 
aly (A +- ba?)”. 
16. fa/a + ba?dx = ( Bp sm) (a ar”) 


Doct pally eens 2ar/a — x 
17. [Fe ae vee ee 


3 
+ 


18. fiat boo") da 


x 


4 
= 37, (a + ba") 


at 
$Y) (Ot bw) 9 arty Gt be) 


2 (a+boy'+a va 
In effecting this integration, transform by assuming 


a be? 2". 


SECTION IL. 


INTEGRATION OF RATIONAL FRACTIONS BY DECOMPOSITION 
INTO PARTIAL FRACTIONS. 


2038. A RATIONAL fraction is of the form 
A+ But Ca*?*+.---+ Ma 
Al Bla Oa? +... Na’ 
in which the numerator and denominator are entire and alge- 
braic functions of x; the co-efficients 4, B..., A’, B’..., being 


constants. 


Denote the numerator of such a fraction by F(x), and the 
denominator by f(x). If the degree, with respect to a, of 
F(x), is not. less than that of /(a~), we may divide F(x) by /(a) 
until we arrive at a remainder of a degree inferior to that of 
f(x). Let g(x) be this remainder, and Q the quotient; then 


F(x a 
a AE. 
J(#) f(#) 
a g(x)dx 
and Qda + = aes 
FER pos J 
As a Qdx can Bae, : found, the integration of the origi- 
d 
nal fraction is reduced to the integration of oy in which 
the degree of g(x) is lower than that of /(x). The integra- 
tion of a is effected by resolving it into a series of more 


simple fractions, called partial fractions; and we will now 


demonstrate the possibility of such resolution in all cases in 
343 


344 INTEGRAL CALCULUS. 


which f(x) can be separated into its factors of the first degree 
in respect to a. 

F(z) 
J (2) 
and that the degree of #’(x) is inferior to that of f(x). If the 
factor « — a enters /(x) p times, we shall have 


J (x) = (% — a)? g (2), 


g(x) denoting the product of the other factors of f(z): whence 


PO 
F@) FG) aa) 9a) 


204. Suppose the fraction to be in its lowest terms, 


7(@) ~ (@—a)?g(@)— (w@ — a) g(a) (% — a)? 


But F(x) — = g(x) = 0 when x =a, and is therefore divisi- 
g(a 
ble by «—a. Let w,(x) be the quotient, then 
2 bee ale Ey ne Bs 
J(")  (@— a)?" g(x) © g(a) (@ — a)? 


Denoting Ae: - by A,, we have 
g(a 


F(x) a w1() A, 


NOMIC er 


F(a) 


F(a) has been resolved into two parts, one of which is 
x 


Be tatico a wi(2) 
ipa? In like manner, (oe ay? lpia) 


w (2) Wy (Z) 4. A, 


(x7 — a)? l(a) (a —a)?—*g(x) ' (2 —ayPal 


and so on, until at last we should have 


may be reduced to 


Wp —1(2) _ Up (#) ou 
(w—a)g(x) g(v) ' e@—a 


RATIONAL FRACTIONS. 345 


By the successive substitution of these values of 


Y1(#) W(x) 
(@—a)P"y(e)' (@ =a) "H@) 
of that in which they were deduced, we find 


in the order the inverse | 


Pm). wh, (2) A, A, 

ee tener Gaett tee 

Proceeding in the same way with the fraction se the 
2 


F(a) 
J() 
completely effected. 


decomposition of 


into partial fractions will be at last 


205. If the root a is imaginary, and equal toa + pv — 1, 
then a, = a — BY — 1 is also a root of the equation /(x) = 0. 
Suppose that all the partial fractions corresponding to the real 
roots have been determined, and that there remains for resolu- 


F(x) 
( 


tion into such fractions the fraction y? in which f(x) =0 
a 


1 
gives rise to imaginary roots alone. Suppose, further, that the 
pair of roots, #—=a+ B/ —1, «=a— pV —1, enters this 
equation g times. Denote the factor of f,(x) that gives the 
remaining imaginary roots by g(x); and, to abridge, make 


Oey — 1, a, — « — b/ — 1: then 


ae f(a) F(a). 
F(a) P(x ) (a)? py(x ) 91 (a) 5 
Ale) @— ore CEO (@—ayl(@— a, 
ae i 
Sie ae’. oie) ( " 


2 (2% 0) G,)% yi epeenit 1} ere ay)! 
44 


346 INTEGRAL .CALCULUS. 
In like manner, 


He nN L itiey 


91 (a) 
Vo (eh 
(eae aa 
ee) — _ PG) F\(a,) — Fa) ay 
ee p1(x) 
eae (4, — @)g1 (41) s 


(x — a)?—*(@ — ay)"9,(@) 


fF (a,) — ao a) 
Sk é (4, — @)g (a) (2). 


—,a)1~"(e%— ay)2 


The last term in the second member of Eq. 2 may be written 
F(a) — Fi(@) 


p1(@1) S g1(@) ‘ 
(a, — a)(% — a)?" (@ — a)? 


But (a, — a) = — 264/—1, and #i (a1) 2) is derived from fac) 
91 (4) (a) 
by changing the sign of / —1: hence, if ae as eae BT. te 
then Fila) a34 > RN dana 
Py (1) 
F(a) & F(a) = Bae 
Pi(41) px (@) 
Therefore 
(ay 1) F(a) B 
91(41) 91 (@) B 


(a, —a)(e@—ay(@ a," (way @— ayy 
and the sum of the two partial fractions in the second mem- 
bers of Eqs. 1 and 2 will become 


RATIONAL FRACTIONS. 347 


B 


BAA 1 B 
a ear 


AL BVITHE (27— 0) BA” sad 


(2 — ae — 1)? 
B 
B (c—a)+A 
— (a? — Qoe a? + Bt 

which is rational. It is also seen, that the numerator and de- 
nominator of the first term in the second member of Eq. 2 is 
divisible by « —a. Dividing, and denoting the numerator of 
the result by %,(x), this term may be put under the form 

X, (x) f Elmina’ <8 
(i — von + 02+) "9,(z) hence, by substitution in Eq. 1, 
we should have 
F(x) 1, (a) = (ea) +4 
F(a) (@— tea pot py "9,(a) 1 Snreanra ye 
Now, %,(a) is a rational and entire function of x, and the frac- 

XL (x) 
(a? — 2ax + a&? + B*)2—! g(x) 


tion may be treated as was 


F(x) 
Si(@)’ | 
of imaginary roots being of the form 
By (a) 2, (a) vei iecapeat 

Ji(@) ~ Oi(%) (2? — 20x a? pet 


and so on; our result with respect to the assumed pair 


| M,« +N, 
| isc® — oe +a? + B? 


The possibility is thus demonstrated of resolving a fraction, 
the terms of which are rational functions of a single variable, 
into a series of rational partial fractions whenever the denom- 


inator of the given fraction can be separated into factors, 


348 _ INTEGRAL. CALCULUS. 


whether real or imaginary, of the first degree with respect to 
the variable. The investigation also shows the form of the 
_ partial fractions answering to the different kinds of factors of 
the denominator of the given fraction. 


Thus if f(x), the denominator of , contains the factors 


F(a) 
J (x)? 
i OR Cb (2 ic), (2 —a— pv —1)?, 
(w—a+pry/—1)?, e—7y—d/—1, w—y + /—1, 
then 


her a A By Bis Nexo 
16) Fe Go, Co a 
a 
(eee a eee ee 
Mx + N, Mia + N, 
+ a Sen pe + Py + GP pon eee 
MU, +N, Pe+Q 


TT ae pop R TT Rep EE 

The labor of determining the constants 4,, A,...B,...M,, 
NV,..., for the partial fractions, which, by following the method 
above indicated, would be very great in many cases, may be 
diminished by expedients which we will now investigate. 
The most obvious of these is based on the consideration, that, 
when the partial fractions are reduced to a common denomina- 
tor, the numerator of the result is identically equal to the nu- 
merator of the given fraction. 

206. To determine the partial fraction corresponding to the 
single real factor, « — a, of f(x). 


Lape (2) 
J(t) &@—a& g(x) 


Assume 


(1); 


RATIONAL FRACTIONS. 349 


in which 4 is a constant, and ae way is the sum of the partial 


fractions answering to the remaining factors of f(a), arid 


J(&) = (% — 4) 9(*). 


From (1) we have 


F(a) = g(a) + (# —a)w(a) (2), 


an identical equation. Make a =a, and then 


eh ei Cea) 
(a) = Ag(a): . Da: Aaah 9 


We also have the identical equation 


J (x) = (& — a)q(a) ; 


whence, by differentiation, 


f' (x) = g(a) + (2 —.a)p’(x), 


an equation also identical: therefore, making x =a, 


J’(4) = (2): ent 


207. To determine the partial fractions corresponding to the 
real factor («— a) repeated n times. 

We now assume 

Peles Ay oa: A; Levin d as w(x) 

f(z) (e@—a)* * (wa) —a ~ 4(#) 
v(x) 


q(2) 
factors of f(a) give rise. 


Multiply both members of (1) by (x — a)", and we have the 
identical equation, 


eae (2) 
g(2) p(X) 
observing that /(x) = g(x)(a—a)". Denoting the first mem- 


(1); 


denoting the sum of the fractions to which the other 


A, + Ay(w—a)+ +++ +A, (a —a)t (2 — a), 


350 INTEGRAL CALCULUS. 


ber of this equation by z(x), and then, in it and its successive 
differential equations, making « = a, we have 
7(a)' = Aj, x/(a) = A,, 7" (0) = oe 
ya) — 1.2.0. (0 ee 

and thus the numerators of the partial fractions are determined. 

208. To find the partial fraction corresponding to a single 
pair of imaginary factors. 

Suppose « — a — Br/— 1, e—a+ fv — 1, to be the ima- 
ginary factors of f(x). We then put 

, F(a) Me +N w(@), 
f(z)” wv —2ax+a?+6 

whence /'(x) = g(x) (Ma + N) + w(x) (x? — 2am + a? + 6?), 
an identical equation. 

Make a=atp/—l; then 

F(a+6¥ —1) = 9(a@+87—1) EAC cc : I) + Nf; 
or, by making « = a— pv — 1, 

F(a —BV—=1) =9(@—pV=1){ Mla —BV=1) + MI}. 
These last equations may be written | 

A+ BV—1=(C+ DV—1)|M(a+pVv7—1) 4+ FI}, 

A— BY—1=(0C—DvV—1)| M(a—bY—1)+ I}, 
in which 4, B, C, and D are known functions of @ and 8. 
From either of these equations, the values of WM and NV may 
be found by equating the real part of one member with the 
real part of the other, and the imaginary part of one member 
with the imaginary part of the other. | 

The values of JM and WN may also be found by the method 
of Art. 206. Thus, for brevity, denote the imaginary factors 


by « — a, x — a,; then the partial fractions are 
Faye Rll (i) i ne 
f(a) &@— a’ f'(a) & a, 


RATIONAL FRACTIONS. 851 


F(a) = sais F(a) my) — . 
If Fila A+ Br —1, then acai ABN since 
F(a;) F(a) 


by changing the sign of /—1: 


Flay) is derived from —— Fila 7 (a) 
hence, replacing a and a, by their values 
a+ bv —1, «— bY —1, 
the fractions become 
etn — Meh ae | 
NS | if er as 
the sum of which is 
rene — a) + 2Bs 
a® — Qo + oc? + B 
209. To find the partial fractions corresponding to a pair 


of imaginary factors which enters the denominator of the given 
fraction several times. 

Let « —a —6V —1, «—a+ $V —1, be the imaginary fac- 
tors, and, to abridge, put a=a-+ b/—1, a, = a— bv —1; 
then, putting f(x) =} (a — a)(a — a,){"9(a), 

Atay, Mix + N, M,«+ N, 
Fe) —Ve—o@—a)f'* [e=ae@=a) 
ue M2 + N,« EDs 
eee ay ys (8 G(s) 
(2) 


9(@) 
which the remaining factors of /(x) give rise. Multiply the 


att 


representing by the sum of the partial fractions to 


first member of this equation by f(x), and the second member 
-by its equal }(#—a)(x —a,)}*% (a), and we have 
F(a) = (Ma + N)9(#) + (Myx + N,)(# — a)(x — ay)9(2) 

+ (Myx + N,)}(@ —a)(e —a)}?9(2) + + 

+} (@ — a)(% — a4){"9(@) (1). 


352» INTEGRAL CALCULUS. 


Now, whether we make a =a, or «=a, all the terms in 
the second member of this equation, except the first term, 
vanish. Suppose «=a, then 


F(a) = (M,a+ ™)9(a) ; 


and if the real parts in the two members of this last be 
equated, and also the imaginary parts, we shall have two 
equations from which to find the values of WJ, and N,. Sub- 
stitute these values in (1), transpose (,a + N,)g(x), and 
divide through by (2 — a)(~ —a,), denoting the first member 
of the resulting equation by F(x); then 
F(x) = (hx + My)q(a) + (Max +N,)(x —a)(a —a,)9(2) ++ 
+ | (@— a)(@— ay)}*~"g(a) (2). 

Proceeding with (2) as we did with (1), the values of M, and 
N, may be found; and, by repeating these operations, all of the 
constants, 1/,, V,, W,, N,..., will finally be determined. 

210. The rational fraction, which may be decomposed into 
partial fractions by the foregoing methods, being a differential 
co-efficient, the resulting fractions are also differential co-effi- 
cients; and the sum of their integrals will be the integral of 
the given fraction. 

The differentials corresponding to these partial fractions are 
of the form 

Adx (Ma + N)da 
(e— a)" (a? + px +9) 

iseger 
m—1 («—a)™—)!? 


becomes Al(~ — a) when m =1; and that of the latter, when 


which 


The integral of the former is — 


nm = 1, has been explained in Ex. 20, Art. 201. The integra- 
tion of the second form, if m be greater than unity, is reserved 
for the next section. 


RATIONAL FRACTIONS. 353 


EXAMPLES. 
(3 —2ax)dx F(x) 
Pie 2 f(a) 
x+1, «—2. We therefore put 


3—2n A A, 
ee er alma os 


The factors of the denominator are 


iE, 


Substituting — land + 2 successively for x in 


Ia) = 3 — 2x 
yA boy) eas 
5 i 
we get 45? A, =— 3° 
(3—2x)dxe 5 da 1 da 
eee ee tA ee Bo 2” 
(3—2x)dx ge Baye be 
——— 210 -+1) 3 (a — 2). 


ie (x? — ae 2)\de 
(@? e+ lf@ +1) 


In the denominator of this, the pair of imaginary factors, 
aa ieee ep t 
Se 3, e+ 5 + 9 4/ — 3, enters twice; and the real 


factor e+ 1 once. We put 

w* — 8a — 2 LPR crates aoe 
(w+ae+l1y(atl) (@+e+1P?° etoe+l 
: xv? — 3x —2 —=(M,x«+ M)(x+1) 
$ Oha + Mart 2+ Weetl) +(e tet Le) (0) 
Give to x one of the values which reduce x? + x+-1 to zero; 


then, for this value, (1) becomes 


x? —32—2=(Myx+ M)(a+1) (2). 


45 


(2) . 
Tel 


354 INTEGRAL CALCULUS. 


From #?-+a2+1=0, we have x? =—a—1. Substituting 
in (2), 
—4¢—3= Me? + Mct+ Noct+M 
=M,(—2-1)+Me+Not+™ 
= — Met Not: 
whence M,—N,=3, N= — 4, A= — 
In (1), replacing M, and N, by the values thus found, and 
transposing, we have 
e*— 384 —2+(x-+ 4)(e +1) = 2(x? + 4+ 1) 
= (Myx + N)(x? +e+1)(@+1)+(a?+a+4+1Py(2) (3). 
Dividing through by x? + «--1, and in the result making 
x?+oaeti=0, we get 
2 = (Mx + N,)( +1): 
whence Mo — 2, 


may be found 


The partial fraction corresponding to aie 
by the method explained in Art. 206; or thus: After dividing 
(3) by «? + «+1, replace M, and N, by their values, trans- 


pose, and again divide by w +a2+1. We find w(a)=2: 


(x* — 3a— 2)dax (a+ 4)dx_ 
therefore fst ap Ihe) = @ aI 
2ada 2dx 


ES i ei 
3. (Seatac 832, aR filsh 
ev*— 5u* + 32+ 9 
By the method of equal roots, we readily discover that the 
denominator may be resolved into the factors (w — 3)?, «+1: 
hence we put 
9x? + 9x — 128 A, A, B, 
gba Lao ay t cS 
whence 
9x? 4+- 9x —128 = 4,(x+1)4+ 4, (x—3)(a+1)+ B(x — 3)’; 


_ RATIONAL FRACTIONS. 355 


from which, by making x=3 and «= —1 successively, we 
get 4,;— —5, By=—8. If the second member were de- 
veloped, the co-efficient of x? would be 4,+ B,; equating 
this with the co-efficient of x? in the first member, we have 


Meee 9: .*. A,—17; and therefore 


{ise + 9x — 128)da _ uO ane 5dac Hef Lia: 8da 
w® —$e?+3a+9 — (7 — gy oh DG ees ae, 


= ae + 171(a —3) — 8l(a+1). 


m—l1 


: e 
211. Integration of ra 


when m and n are positive 
integers. 


If n be an even number, the real roots of «” —1=0 are 
+ 1 and — 1; and the imaginary roots (Art. T7) are given by 


the expression cos. Wea 1 sin. a , by giving to & in 


succession the values 1, 2, 3..., i ifr 
We will denote the arc — = by A, ae - being the fraction to 


to be resolved into Aa fractions. It has been shown 
(Art. 206), that, if a be a root of /(x#) = 0, the corresponding 

m—t\ 
per i : hence, for the fraction —~ 


f(a ) sy ’ ar — 1? 
the partial fraction for the root Nie 1 of the equation x” —1=0 


partial fraction is 


m—1 m 
Fy oo ee ea ee el and, for the root — 1, the partial 
na"—* na” n(x —1) 
so" 
n(aw + 1) 
cos. 2k + / —1, sin. 2k0, 


give the partial fractions 


fraction is The pair of imaginary roots, 


356 INTEGRAL CALCULUS. 


(cos. 2k0-+ / — 1 sin. 2k9)” 

n(x — cos. 2k0 — +/ — sin. 2h) 

4. _ (cos. 2k6 — /—1 sin. 2k0)™ 
n(a — cos. 2k0 + 4/—I1sin. 20)? 


that is, (Art. 73), 


cos. 2mko +- / — 1 sin. 2mk6 
n(a — cos. 2k9 — 4/ —1 sin. 2k0) 
cos. 2mk9 —4/—1 sin. 2mko 
n(a — cos. 2k6 + 4/—1 sin. 2k9) 
__ 2cos. 2mk#(a — cos. 2k4) — 2sin. 2mké sin. 2k0 . 
i n(x? — 2acos. 2k6 + 1) ; 
and for each pair of imaginary roots, that is, for each of the 


values 1, 2,3..., er — 1 of k, there will be a partial fraction of 


the form of this. Let the symbol S denote the sum of these ; 
then 


a ¢ wn Lae 
Sata —1) +f pate 


ahs eal cos. 2mk6(a — cos. 2k0) — sin. 2mk6 sin. 2k0 
(~ — cos. 2k0)? + sin.? 2k 


=2Ka— 1) je sPi(e+1) 


+ bs cos. 2mk6l(x? — 2x cos. 2kd +- 1) 


_,«% — cos. 2k0 
sin. 2k0 


by observing that the last term under the sign of integration 


2 : 
ed S sin. 2mké tan. 


can be separated into the two fractions 


2 _cos. ee — cos, 2k) ie 2 sin. 2mk9 sin. 2k 
n~ — 2x cos. 2k9 +1 “Hay Die a ame (2 — cos. 2k9)? + sin? 2k9 
m—1 
212. Integration of es mand n being positive inte- 


gers, and n an odd number. 


EXAMPLES. Se 


In this case, «* — 1 = 0 has but one real root, +1; and the 
imaginary roots are the values assumed by the expression 


COS. eT 1 sin. ais by giving to & in succession the 


palder lt 2, 3... 


Hence, by operating as in 
the ae article, we find 


am 1 
capes we —1) sh ly Cos. miket(a — 2x cos. 2k0 + 1) 


bag he — l 
_1 © — COS. 2h9. 
sin. 2k9 


ae 
= 5 sin. 2nké tan. 
nN 


gm =! 
213. Integration of ~ ey ~__°" m and n being entire and 
positive, and n even. 
Under the supposition, none of the roots of #* + 1=0 are 


real; and the ee roots are found by giving to k, in the 


a nee . te/—1 Ane zx, the values 


expression Cos. ? 


Oe a: te lin succession. Put 6 for , then the partial 


fractions corresponding to a pair of these roots will be 
cos. (24 + 1)4 + /—1 sin. (2k + 1)4 
aw — cos.(2k + 1)6 — WV —1sin.(2k + 1)0 
cos. (2h + 1)9 — /—1 sin. (2k + 1)9 
@ — cos. (2k +1)0 ++/—1 sin. (2k + 1)9' 
the sum of which is 
2 cos. m(2k-+-1)@ } x — cos. (2k-+- 1)9 { — sin. m(2k +- 1)@ sin. (2k +- 1)0 


tt | 2 — Cos. (2k 1)0{" +L sin.? (2k-+ 1)0 
Hence 
Tyla 1 1 
| eee —~= Scos.m(2k-+ 1)0l | @ — 2xcos.(2k-+1)0-+1} 
x — cos.(2k + 1)0 
sin. (2k + 1)9 


+. : * sin.m(2k + 1)#tan.—! 


358 INTEGRAL CALCULUS. 
m—l 0 
ated) ? 


finding the partial fractions corresponding to the roots of 


In like manner, we integrate when n is odd, by 
x” +-1=0. In this case, there is one real root, — 1; and the 
other roots, which are imaginary, are the values assumed by 
the expression | 

cos. (2k + 1) = at /— Tain (2a = 


n— 


by giving to & the values 1.2.3... successively. 


We should find 


m—I — ym 
f= ety) 


a 


es aoe +1)0 a” — 2a cos.(2k +1)0+-1 


oe a sin.m(2k + 1)étan.—! % — cos. (2h + 1)6- 


sin.(2k + 1)0 
EXAMPLES. 
y las ees 3 qin Soe 
Sate e ep 
44) eee cen mars Oa iT tn") 
f Nee ee 
6. [eS = alle yee gee’ Be) 


a ; /3 | tan.—\(2e — 4/3) — tan.—"(Qa + 4/8) . 


EXAMPLES. 359 


ig J ae. This may be rationalized by putting 


(1 — a)” 
Fast etoc? 2°: 
2 1 
whence da=— Beis (1 — AN es ea 
(2° + 1)° (1 + 23)° 
dx 2dz 
Ca eat rr 
gu wae 1 _,22—1. 


htt pe 
in which, if we replace z by its value ee we have the 


required integral in terms of a. 


SECUIO Me 


FORMULA FOR THE INTEGRATION OF BINOMIAL DIFFERENTIALS 
BY SUCCESSIVE REDUCTION, 


214. Tue integration of differentials of the form 
x(a + ba”)? dx 

may be made to depend on that of other expressions of the 
same form, in which the exponent of the variable without the 
parenthesis, or the exponent of the parenthesis itself, is less 
than in the original expression. This is accomplished by the 
method of integration by parts. We have 

fen(a + bx")?dx sap agin (a+ bx”)? a—dax = fudv 


Pat or facia MRL oe 
by making u = x a pean 


and therefore 
am mit % ee m—n—-] (4+ Oras 
sfx (a+ bx")?dx=«x Aa abl, 
Le pearie Ma RE Se agent 
She Sane fe (a-+ ba?) Phd fT). 
The integration of «”(a-+ ba")? is thus brought to that of 
a™—"(a-+ bx”)? +!, which last is more simple than the first 


when m is positive, and greater than n, and when p is negative; 
for then the numerical value of p + 1 is less than p. 

But we may find a formula in which the exponent of the 
variable without the parenthesis is diminished, while that of 
the parenthesis itself is unchanged. 

Thus we have the identical equation 

™ "(a + bar)? Th = o™—"(a + ba”)? (a + ba) 

= ax™—"(a + bax")? + ba™a + ba”)?; 


360 


REDUCTION FORMULA. 361 


therefore 


fen—"(a + bx")? *1dxe = afam—r(a + bx")? da 
| +b fa™(a + bx")? da. 
Substituting this value in Eq. 1, we have 
ber yrr 
m ba")? dr = EAS 
fa (a+ bx")?dx= a CBee TS 
m—n+1 gee % 
Cea (a + ba”)? dx 
ea eae y — b fx (a+ bx")?dx; 


whence, by transposition and reduction, 
ema tlg + ban)Ptt 
b(m + np + 1) 
a(m —n +1) 
~ O(m + np + 1) 
The integration of «(a + bx")? dx is then made to depend 


fea + ba")? dx = 


eae + bx")?dx (A). 


on that of «”~"(a + bx”)?dx; and, by another application of 
the formula, the integration of this last reduces to that of 
am— (a + bx")?dx, and so on: hence, if m is positive, and 


greater than n, and ¢ denote the entire part of the quotient ~ 


the integral to be determined after a number, 7, of reductions, 
would be 
a enn + ba")? dx. 
If m— im =n — 1, this expression is immediately integra- 


ble; for 
Bet Oe") PEE 
nb(p +1) ’ 


—21+1, andthe condition 


ernie + bx")? dx = ( 
m+1 
n 


but m — in = n — 1 leads to 


of integrability (Art. 202) is then satisfied. 
46 


362 INTEGRAL CALCULUS. 


Formula A cannot be applied when m+ np +1=0; for 
then its second member takes the form o — oo: but in this 


m+1 
nN 


case + p is equal to zero, that is, an entire number; and 


the original expression is therefore immediately integrable. 

215. Formula for the reduction of the exponent of the 
parenthesis. 

Assume 

x™(a + bx”)? dx = (a+ bu")?d an = Udve 
then, integrating by parts, 
e™tl(a, + ba")? 
m+ 1 


— Bi fem (at bat)? de S411). 


fora + bx")? dz = 


In this formula, the exponent of the binomial has been 
diminished by 1, while that of « without the parenthesis has 
been increased » units. We may, however, diminish the for- 
mer without increasing the latter exponent. In formula A, 
last article, change m intom-+n, and p into p — 1: we thus 


have 
m+n n\p—l wha, am that be")? 
fa +"(a + bx”) We hate aes 
(m + 1)a m n\p—l/-. 
~ tne Ean (a+ bx")? di 


and this value of fa™+t" (a + bx")?~! dx, substituted in Eq. 1, 
gives, after reduction, 
eta + bx”)? 

np + m-- 1 


tet coe fen (a + ba")? dn (1 ae 


fan(a + bx”)?dx = 


REDUCTION. FORMULZ. 363 


By the repeated application of this formula, the exponent p 
will be diminished by all the units it contains. This formula 
will not admit of application when np + m-+1=0; but then 
the integral fa™(a + ba”)? da can be found at once (Art. 202). 
By means of formule A and B, the integral fa™(a + ba")?da, 
when m and » are positive, may be made to depend on the 
more simple integral fa™—(a-+ bu")?—%dx ; in being the 
greatest multiple of m less than m, and q the entire part of p. 

216. Formula for the reduction of the exponent m when 
m is negative. 

From Hq. A, Art. 214, by transposition and division, we find 
am—ntl(q, + ban)P+! 


eee + bx”)? dx = 


a(m —n-+ 1) 
b(m + mp +1) Fm np 
salt aeneneist a fa (a + ba”)? da. 


Changing m — n into — m, this becomes 


ee ee ba® Pk 
faa + bx”)? dx = — ae 1) ei 


bin bc 1 
ite eet ) fo-m-+n(a 4 bar)” dex (C). 


If in denote the greatest multiple of contained in m, then, 
by 7+ 1 applications of this formula, the integration of 
a—™(a + bx”)? dx 
will depend on that of a~™*¢FP"(q 4+ bx")? dx; and, if we 
have —m+(1+1)n=n—1, 


Gx" )\2 ei 
we have era -- ba")? da = (a + ba oy 
But under this supposition, since hae = — 1, an entire 


number, the original expression is immediately integrable 
(Art. 202). 


364 INTEGRAL CALCULUS. 


217. Formula for the reduction of the exponent p when p 
is negative. 
From Formula B (Art. 215), we find 
e™*l(a + ba”)? 
anp 


fora + bx”)? dz = — 


| i 
at ae fon(a + bu")? dz; 


and, if in this we replace p — 1 by — p, it becomes 
| POM aS Oa rras 


fara + tory-ran = 2 eee 


1 me 
age fee 1 Jena +- bu")-? Fda {D). 


By the continued application of this formula, the exponent 
of the binomial will finally be reduced to a positive proper 
fraction. When p=—1, it cannot be applied; but then the 
integration of the given expression may be brought to that of 
a rational fraction. 

218. The preceding formule facilitate the integration of 
binomial differentials; but it is to be observed that the exam- 
ples to which they are applicable belong to cases of integra- 
bility before established (Art. 202), and the results may there- 
fore be obtained independently. 


By the application of Formula A, we have 


1 (ee 28 0 UT ek on Bae 
Bes ate mn a 
and, by making m = 1.3.5... successively, this gives 
ada : 
Wit a) 
x* da aA 2 xd 


REDUCTION FORMULZ: 365 


whence 

ada 

Ae orase 
V1 — a? ‘ 
a da ac? 

=a, 

Vie? (s+ Opa - 
ada act 


When m is an odd number, we have the general formula 
a™ da 

J 1 — 2? 

i gam—l (m —1)a™— —3 het) Se 

= m 45 (m — 2)m a +5 ‘ (vi=a 


and, when m is an even number, 


ve 
i 


tee (1 )ae Leo Bt (iL) ee ee 
. Ce om en av T=@ 
1.3.5...(m—1).. _, 
a eiieeeer Uy ky ero Ona 
2. {——_. ~= fa eae 8 4g?) 3 de, 
a +. x)? 
Comparing this with Formula C, and making 
1 
ese, te re 6: 
we find 3 
de (at tat) 
ad aa x)? (m — 1)a? gmt 


‘Auta f—"—,; 
(m—1)a?* «2™—?(a? + x)? 


366 INTEGRAL CALCULUS. . 


Without referring to the formula, this expression may be found 
as follows : — 


2 2 
f dix (fete ad(a? +- x*)” My 1 1s 
x™ (a? 4 2)? rs rae go a 
2 2 
_(a+2%) +(m+1)f—“~+" _ ae dx ; 
7 oe omg? 4 a2)? 
whence, by transposition and reduction, 
1 
die a? + a)? d 
(m + yar f— “2 _, EE) _ 4, f__S _, 
ere dats Cs fe + a*)* gam +l Aehdee (e 2 NG 
ii dx See Ce x?) 
e™(a? + 2)? (m—1l)a?a™! 
m — 2 dx 


(m — 1)a? oq? 4 gp) 


. (Mx+N)dx 
219. By means of Formula D, the expression (oT cee ee 


which occurs in Art. 210, may be integrated by successive 
reduction. 


Let a+ ba/—1, «—B/—1, be the roots of the equation 
2? +pxe+q—0; then 
(Mx + N)dax (Mx + N )dax 


(a? + pe +9)" | (a — a)? + 6?}" 


_ Mle ade Dy i yo 
[e—aptey" | 3 Te a By? 
Putting « — a@=2, we find 
M(a—a)de ‘ Mee 
Steep + ep 5 ae 
Bl 5 1 
ib2@.— 1) (ee 
M tl 


REDUCTION FORMULZ. 367 


Making « —a=y, and Ma + N= WM’, we have 
fe Meck Mis f MH parry 
and therefore, by combining these results, we have 
f (Mea +- NV) dic 2 M 1 
| (a — a)? + B?}” 2(n—1) {(w@—a)+ ptr 
+f" (y? + By "dy. 


Formula D may now be applied to the term under the sign ‘f 


in the second member of this last equation; and, by repeated 
applications, the exponent — n will be reduced to — 1, when 
the integral will be completely determined. 

220. Reduction formule may also be constructed to facili- 
tate the integration of trigonometrical functions. Let the 
integral of sin.?a cos.’adzx be required. 


Make sin.2 = z; then . 
BK paint 
cos. a = (1 — 27)", de = (1—2?) “dz, 


q—l 
and sin.?xcos.%xda = 2?(1 — 2”) ? dz. 


Now, if g be an odd number, whether positive or negative, 


we may always effect the integration of z?(1 — ay ae, what- 
ever may be the value of ». In like manner, by making 
cos. = 2, we see that the integration can be effected when p 
is an odd number, whether positive or negative, whatever be 


the value of q 


eee! 


Formule A and B, we get 
yP— 1 — 27) 2 


q—l 
UA EN ty 2 Je 
fe Z*) 2 dz ar 


25h fera—ay Fide (a) 


368 INTEGRAL CALCULUS. 
fea a wy? de = Seager) 28 ps 
(mae | 
—3 
Beene. =f e(1 — 2?) © de | (B’). 


wach g 
If p < 0, Formula C gives 


g—l Pi ue 

vo = 5 Fan 

fe? — 2°) 2 d= — AL ld 
p—l 


== ; = 
+PoI—*fa-7411 9!) Fda (0%); 
and when a Wee 0, by For mula D, we have ) 
fou ortMyata—e" 


P| ee 2) F de (D). 


By the aid of the foregoing formulz, we are enabled to make 


the integration of sin.?x~cos.%adx depend on that of expres- 
sions in which the exponents p and q are numerically less than 
in the original function. From (A’) we have 


fosin Px cos.2ada = sin.? “'wcos.? Fix 


Pd 
4+ PEA fsine-tecos.tad (1); 


and from (B’), 
sin.? tlycos.2—!a 


Prd 
+i fsin2 cos.?—"ada. (2). 


sine cos.!ada — 


When q is positive, by the application of (2), 
fsin.?a cos. fada 


will finally depend on fsin.?adx, or on fsin.?# cos. xda, 


according as q is even or odd. 


REDUCTION FORMULZ. 369 
By making g = 0, in (1), we have 


ag et 
: sin.?—*xcos.« , p—I1e. 
J sin.2xde = Jsin.? ade; 


L 


and thus, if p be a positive integer, and even, fsin.? ada will 


at last depend on bh: dza—x; andif pbea positive integer, and 
odd, the final integral to be found is fsin.wda = — cos.” In 
the second case, that is, when g is odd, we have 


sin.? tly 
pri 
It is therefore always possible to find the integral 


J sin.? cos. ade = fsin.?adsin.« = 


of sin.” acos.? ada 


when p and q are entire and positive. 
Formule 1 and 2 are inapplicable when p= — q; but in 
this case 


fi sin.? a cos.%ada = f tan.? dx = fh tan.?—2a tan2ada 
= ftan.?~* «(seca — 1)dax 


= ftan.?—? ad tan. x — ftan.? ada 
tan? <x Be 
ee aon — fian.?-*adx. (8). 
This formula serves for the reduction of the exponent of 
tan.«; and the integration will at last depend on that of 


fdx =a, or ftan.xdxz =1cos.a, 


according as p is even or odd. 


221. In (1) of the preceding article, making g=0, we 
have 
eee ie 
fsinrade peep eet Epes COR. 4+? 7 fein rade; 


a! 
47 


370 INTEGRAL CALCULUS. 


and hence, when p is even, 


; COs. @ 1 
sin.? ada —= — sin.? —!a aay = 9 SID. P—3o 
=: 


pipe ae 3 
t (p—2)\(7 =) 
Cp Th a 

(p — 2)(p —4)...4.2 

(p —1)(p —8)...3.1 & 
+ (eae rast (1); 


sin. x 


and, when » is odd, 
J sin.? ada = — 


COS. @&@ 


sin? -|- t = 5 sin.?—8e oleipiee 
(p — 1)(p —3)...2 2) 
(p —2)(p — 4)...1 
In like manner, by making » = 0 in (2) of the preceding 
article, we should have formule for 


fcos.%ada. 
EXAMPLES. 

1 

1 otda 9... a°Td( 2am ae )a 

e iY Ma aaper eaecets & ae tl eg a ae a a 
; (Qac—a*)° - n 

2n —1 bee 2 8b 
= af a 
n (2aa — sx)” 


This will finally lead to 
ih da <0 Seo 
mromcr mre. | ver. SIN. r= ¢ 


(2ax — x?)? 
xe” ox 
2a eae : 
(2aa + 2”) 
_. @\( 2am + x2)? - 2 ah ada 
n Ls eae (2aa + x)* ; 


EXAMPLES. 371 
and ultimately we should have (Ex. 7, p. 328) 
ah oan 1 a3) 
(2aa +- a”)? 


x(a? pi aye 


3. f@ = 2*)"da = Pia 


whe ae a? | (a? — «*\"—"da. 


If n be a fraction belonging to the series 4 ae this 


process of reduction would at last lead to 


LE LACE Ss 
_________ == sin. !-. 


(a? a Zi). a 


3 a 2 
2 2 Soe it 2 
4, f(a a?) "da = 7 (a oo) 
Se rar Aa ee 
an x(a — x”) ++ -g sn. rt 


5, f(@+e'yde= Oar a IT 7 a? | (a+ rh anes 7: 


When vn is any one of the fractions 7 = Bree the integral 


2 


bo 
ho 


will finally depend on 


bole 
“— 
es 


f—S Hlet (+0 
(a? + a)? 
da 1 i 
: S@—ay ae @—ay 


2r—3 1 da 
+r 2(n — 1) a? Sie — gijet 
: ee T 2 
When 7 is one of the series of fractions a? 9 97” the in- 


tegration will finally depend on that of 


372 INTEGRAL CALCULUS. 


oe x 


2 Se 2% ok 
(a* — 2”) a?(a? — a”) 


by Formula D. 


da i x 2 x 
[#1 te 
(a?— a)? 3a? (a? —@’)? 8a? aa? — o) 
1 
8. (jee =_lc a Caeser 
a"(a® — x)? (n — 1)a? ar! 
| peared det 


m— Liga one (ae dies x)? 


When n is even, the application of the formula will lead to 


f dx (a — ay 
x(a? — 2)? a*a 


and, when n is odd, the integral will depend on that of - 


1 
jy a 
a a+ (a? —2%) a x 


ak 
x(a? _—_ iden pe 


(Ex. 17, p. 332). 


P} ) 

9 if ade  3”"(a+ bx)"  — 8na anda 

(abe)? <(8n+2)b  3n+2b” (a bx)? 

By the application of this formula, the exponent n will at 
last be reduced to zero, and the integration will depend on 
that of 

da 3 5 
(a+bx)* 2b 
2 2 2 : 2 
10,08 Gof ee = (a+ bay! | a 
(a+ ba)? 


8b 206? 400? 


EXAMPLES. 3:10 


(a+b+ cx)? NC 
Patt eet ef CTNas Dh? HS 


Hy (a+ ba + ca? iy 


i f a” die xe"—*(a + bx + ay 


on—1B eS) Ly : 
oi oS RO its eS. nee 
2n c (atbet cx)? 
and, ultimately, we shall have to find 
f xd _ (a+ be + ae b da ; 
(a + bx + cx)? C 2c (a+ bet ca?)? 
but the integration of Reamer oe has been explained 
(a+ bx + ca?)® 
(Ex. 7, p. 328). 
2 
Pee BS (9) 9g 1 'g2)? 
(2 — 2x + «*)* 2 


+5la—1+ (20420) 
13. fa(2ax Pet) eles — 5 (2ax — a2)! 


+ af (2ax — x) de. 

2a Bf na, 

14. J, a( 2ace — x*)*da = cn 
[ @*(2ae — x") dx = ei (2ax — x*) 


are ne a(2aa — x?) dx. 


Sat 
i: 2ax — x? *g 
fo n?(2a0 — x7) dx = ai 


di sin. I 11 — gin. aw 
17. sin.? a 
i - mee cos.” Bereta 4s 1+ sin.x 


SECTION IV. 


GEOMETRICAL SIGNIFICATION AND PROPERTIES OF DEFINITE INTE- 
GRALS. — ANOTHER DEMONSTRATION OF TAYLOR’S THEOREM. — 
DEFINITE INTEGRALS IN WHICH ONE OF THE LIMITS BECOMES 
INFINITE. — DEFINITE INTEGRALS IN WHICH THE FUNCTION UN- 
DER THE sien f BECOMES INFINITE. — DEFINITE INTEGRALS THAT 


BECOME INDETERMINATE. — INTEGRATION BY SERIES. 


222. AssumE CPD to be the curve of which the equation, 
when referred to the rectangu- 
lar axes Ox, -Oy, 181 y =A a). 
It has been shown (Art. 164) 
that /(a)dx is the differential 
of the area of a segment of 


the curve terminated by a va- 


riable ordinate ; and therefore 
J7(x)d« is to be regarded as 
the expression for the area bounded by the curve, the axis of 
abscisse, and any two ordinates whatever. If this integral 
be taken between assigned values, a and 8, for x, the area will 
be limited in the direction of the axis of x by the ordinates cor- 
responding to these values of x But the arbitrary constant 
may be determined by the condition that the area shall be 
nothing for « = OA =a; that is, shall be limited on one side 
by the fixed ordinate AC, while on the other side it is bounded 
by a variable ordinate corresponding to the variable abscissa 


OM =z; 
874 


DEFINITE INTEGRALS. 375 


If [f(x)de = w(x) + CO, then, by the above condition, we 
should have | 
yw(a) +C=0, C= —y(a); 


and 
[7 (ada = y (x) — p(a) 
is the expression for an indefinite area taking its origin from 
the fixed ordinate AC. 
When the other value of «, «= OM’ =, is assigned, the 
integral becomes definite, and we have 


f A@dx =) — va). 
Definite integrals, when applied in the determination of the 
length of curves, the surfaces and volumes of solids, admit of 
a like interpretation. 

223. In Art. 192, it was shown that a definite integral was 
to be regarded as the limit of the sum of an infinite number 
of very small terms. 

To illustrate this proposition geometrically, let y= /(x) be 
the equation of the curve CMD referred to rectangular co- 
ordinate axes, and suppose that, between the assumed limits 
aa, «—b), y increases continuously. 

Then /(x)Aa@ measures the area 
of one of the small rectangles 7% m?—D 
MP’, M' P"...; and if 2/(x) ax de- ya 
notes the sum of all these rectan- y, 
gles, the area ACDB included be- C 
tween the curve, the axis of a, and 
the ordinates y=/(a), y=/(d), 
will be expressed by 


lim, 2/(w)ae= f f(w)de = vb) — (a), 


376 INTEGRAL -CALCULUS. 


if w(x) be the function of which /(#) is the differential co- 
efficient. 

224. The order in which the limits of a definite integral 
are taken may be inverted, provided the sign of the result be 
changed for 


[ f@)te= 90) — 9) 
J'Fl@)dz= va) — v@): 


if f(a)dx = — if i f(a)dx. 


Also, if ¢ be a value of w intermediate to the limits a and 8, 
we have 


J F@)dz = v(0) — ¥(a), 
f f@)te=¥0) —¥(0); 
f Sod = 0(b) — 9(a); 


f faa ag J f@)az +f f(e)de: 


and generally, if there be any number of values ¢, c’, c”..., 
between the values a and 0, it may be shown that 


f s@ae == J f@de +f s(@)de oo 3e Poof mae. 


aos 


ax5d. Let f(x), p(x), be two functions of x, so related that 
J(x) > g(x) for all values of x from x=a to x=); then, 
taking f(x) — g(x) for the differential co-efficient of another 
function, we should have 


[\F@ — 92) } de >0; 


DEFINITE INTEGRALS. 377 


since the derivative f(x) — g(a) is constantly positive be- 


tween the limits a and 0, and the function f'} /() ~- a (2)} de 


is an increasing function of 2: hence 


f fo)ae > f 9 @)dx, 


Also if g,(«) is another function of x, such that g,(x) > /(x), 
for all values of « between the limits a and b, we should have 


f f@)dx< 1 pi(x)dx: 


therefore 
fo ride > fPfw)dx > f* 9(w)de. 


When a given differential cannot be integrated, it is desira- 
ble, and sometimes possible, to find two other integrals be- 
tween which the required integral, at assigned limits, will be 


included. 
EXAMPLE. J 7 sae For values of « between 0 and 
(1-2)! 
1, we have 
l< 1 1 


TTR TE VT ee age cmen yg 
(Lien (1 —'x?)* 


|S ewe oA ras 


a—ayt 46 aa} 
0.5< il = 0.5236. 


bho] 


dic Ants 
——— < sin. 
(L— x?)" 

226. Demonstration of Taylor’s Theorem dependent on 
the properties of definite integrals. 

The equation 


f(e+h) f(a) = ff @+h— oat 


48 


378 INTEGRAL CALCULUS. 


is identically true; but successive integration by parts gives 
[i P@th—Odt=o (@+h—N+f Yr (@+h—ddt, 
fi tf! (a +h —t)dt = © Pet h —t) 
S, Seth — t)dt, 
lies se (w+ h—t)dt= 75-5 f"(w@+h—t) 


“f agli (eth— tat 


‘ n—1 
firma th id= Pe thee 


+f,i 


Making ¢=/ in these equations, and then adding them 


— f(a +h — tht. 


member to member, we have 


J (@ +h) er wae 


1 
rats f.nmee 


If the function to be expanded, and also its differential co- 
efficients up to the order denoted by n+1, are finite, and 
continuous between the limits x and w# +h, the residual term 


1 h 
(n +1) as ‘4 
oes ifs Sf (7 + h —t)dt may be replaced by 
i (n+) ume 
2, ae (tO 


and the expansion then agrees with that of Art. 61. 


DEFINITE INTEGRALS. 379 


227. In what precedes, it has been supposed that the limits 
a and b of the definite integral sf ; J (x)dx were finite, and that 


the function /(x) was also finite, and continuous between 
these same limits. It may happen that one of the limits, 8, 
becomes infinite while the other is finite, the function re- 
maining finite and continuous. Then the value of the inte- 


gral is the limit of the value of OL: when 0 is increased 


without limit. This value may be finite, infinite, or indeter- 


minate. 
EXAMPLE 1. fede. 
0 


For the indefinite integral, we have 
fede =—e "+0: 
2 fi erde=i—S, 
0 
foetde=1 ee 1 
: € 


FIX. 2: ff, era. 


The indefinite integral is 
ferda=er+ a: 


[ e* dan 67 1) — 0. 
0 


380 INTEGRAL CALCULUS. 
Hix. 4. f° cos. ade 
0 


In this case, fcos.cdx = sin.a + C; and, taking the inte- 


gral between 0 and the finite limit 0, we have 


b : 
J cos.“2dxe = sin.b; 


0 
but, when 6 becomes infinite, the value of sin.b will be inde- 
terminate, though confined within the limits 0 and 1. 

The following investigation will sometimes enable us to 


decide whether the definite integral i f J (x)dx is finite or 


infinite forb=co orb=—o. 
First suppose that 6 is very great, but not infinite, and let 
ec be a number comprised between a and 6; then (Art. 224) 


b c b 
[ fwjde =f" f(a)de + f f(w)de. 
Since /(x) is finite, [" f(w)de is also finite; and it remains 


only to examine the value of f f@de when 6 becomes in- 


finite. 


fee (2) Ze) g(x) being a function that remains finite 


for all values of x greater than c. If A denotes the greatest 
and B the least of the values of g(a) for all values of x greater 
than c, we shall have 


A BY 
b od 
J F@de ele 


b A 1 1 
oF freee <5 (si- 5) 


DEFINITE INTEGRALS. 381. 


Now, when n > 1, the second member of this last inequality 


A 1 : 
Cteytte 1: hence, in this case, we 


for b=oo reduces to 


know that the integral if “f(a)de has a finite value. When ; 


n <1, we have 


ig fi(ajdx.> Bic ae 


ave [fod Pe we Gi —c}—"), 

Now, when 1 — n > 0, the second member of this inequality 
becomes infinite for b =o: hence f f@ae, and therefore 
f ° f(x)dx is infinite for b= 0 . 

Te m = 1, then 


['serde > B'S = wi; 


but (2) =—oo whenbU=o: hence f J (a)da =a’. 


Putting f(x) under the form Bee g(x) being a function 
that is finite for all values of « between — o and some value 
less than 0, it may be shown in like manner that {i ° J (a)da 


is finite if m > 1, and infinite ifn << lorn=1. 
Thus, if it be possible to put f(x) under the form ue) and 


the condition imposed on g(a) be satisfied, we can decide 
whether the integral if : J (x)dx is finite or infinite when one 


of its limits becomes + 0 or —o. 
228. Definite integrals in which the function under the 
sien of integration becomes infinite between or at the limits. 


382 INTEGRAL CALCULUS. 


The function /(x) may become infinite at one of the limits, 


b, of the integral if : j(x)dx; in which case the integral is 


defined as the limit of ff (w)de when @ is decreased with- 
out limit. In like manner, if /(«) becomes infinite for «=a, 

6 aie b ae 
then 5 J (x)dx is the limit of ff (ade when « is indefi- 
nitely decreased. Finally, if /(c) =, e being comprised 
between a and b, we should have 

b : c—O 3 b 

f f@de= lim. fi J (x)de + lim. ff (@)am, 


when @ and £ are decreased without limit. Should there be 
more than one value of x for which /(x) becomes infinite be- 
tween the limits a and 0, we learn from what precedes how 


to define the integral ff (a)dz. 
229. It may sometimes be decided whether the integral 
if : J (a)dx is finite or infinite when /(x) is infinite at one of 


the limits. Suppose /(b) = oo, and let f(x) = (Ne: ua eae gy (x) 


being finite for « = b and for all values of x < 0b, and n being 
Ps), 


If ¢ be a number comprised between a and 3b, we have 
b c 
f f@de= [fade + fo sade. 
Now f° f(@)de is finite; and hence fF (2)de will be finite 


or infinite according as if s J/(x)dz is finite or infinite. 


Denote by A the greatest and by B the least of the values 
of g(x) for values of w included between c and’. If n < 1, 


DEFINITE INTEGRALS. 383 


we shall have, for such values of x, f(x) < esis ty and 


(6 — a)”’ 


therefore 


fo re@de < f= Geo a}. 


When @ converges towards 0, the second member of this in- 


A 
faa! a Came 


equality converges towards the finite value 
hence, in this case, the value of lim. Hi oy J (x)dx, and there- 
5 a eye : 
fore of [ J (x)dzx, is finite. 
But if n 4 1, the proposed integral is infinite; for, since 


I(2) >= i= yi we have 


b- Bdx B 1 1 
if “f(a)de ee iis (6 — a) prede TE | Hee me eat 
and it is evident, that, when « becomes 0, the second member 
of this inequality becomes infinite: hence, under the supposi- 
tion, f F(@)ae is infinite. 
In like manner, when n = 1, we have f(x) > en and 
therefore 
van b— 06 cso b—c 
de * f(x) da > i} a - 
b 


ns: 5 ‘ ; 
But soem becomes infinite when a vanishes: hence 


f f@)de, and therefore f Sade, is then infinite. 


EXAMPLE 1. i : ETE — 
a — 0X — X” 


P being a function of « that remains finite for all finite 


384 INTEGRAL CALCULUS. 


values of x, and a and b being two positive quantities, we 


have . | 
Py fee ey P 1... “Spee 
ee Nt ee 
by putting Soa a7) 


Since the exponent of 6 — « is less than 1, it follows from 
the rule just established that the proposed integral has a finite 


value. 
1 dz dx 
Xe ee —=——=* We have 9) s-ccee ee 
vice We have | 7 
. 1a 00. 0 i/o: 
0 / 1 — 2 MEN ice: 


is the differential of the area includ- 


ae 
V/1—ax 


da 
The expression —- 


V/1—x 


ed between the axis of x, and the curve having y¥ = 


for its equation. 

This curve has two asymptotes; the one the axis of a, 
and the other a parallel to the axis of y, and at the distance 
+1 from it. It is Be from 


the figure the f ae rep- 


resents the area bounded by AQ, 
AB, the curve, and its asymptote 
BD; and this area, although it 
extends indefinitely in the direc- 
tion of the asymptote BD, still 


has the finite value 2 


230. A definite integral may become indeterminate, as is 


the case for 


co . 
if sin. xda == cos. 0 — cos. 0, 
0 


INTEGRATION BY SERIES. 385 


since cos.2, when «& is indefinitely increased, does not con- 


verge towards any determinate limit. 


For another example, take J sis ee , In which a and 6 are 


—a 


ie 1 : 
any two positive quantities whatever. Since — becomes in- 
x 


finite for the value x= 0, which is comprised between — a 


and +b, we put 
ee 3 +o dx. 
fo gam. fod tim. —; 


+8 2 
a and $ being numbers numerically less than a and 6 respec- 
tively, and the limits indicated being those answering to 


Gee tee. “But 


[i gsta—t, ees 1B 
x iat 


ath, eae bts, )+2(5). 


Therefore tie ee — i) + lim. (5): 


ON > he : 
The first term (7) in the value of this integral is deter- 
minate; but, since the variables @ and £ are entirely inde- 
pendent of each other, the term lim. (5) does not converge 


towards any fixed limit, and the integral is therefore indeter- 
minate. 
231. When the integral of Xdz is required, and X can be 
developed into a converging series, 
ASUytt,fust-+u,+7, (1), 
we shall have, after multiplying by da, and integrating between 
the limits a and 6, 


iE Xdo= fi wey dar +f wadart + fe, d+ fr r,dx (2). 


49 


386 INTEGRAL CALCULUS. 


If series (1) is converging for «=a, « =), and also for 
all values of x between a and b, we may assume r, < @; 
a being less than any assigned quantity when n is taken suf- 
ficiently great. Whence 


fo rade < fi ada, or f r,dx < a(b—a). 


Therefore } . y,dx will decrease without limit when 7% is in- 


a 


creased without limit; whence the series 
b Baie b 
i} wdc + f U,Axe +t... Ete U,ae 
a a a 


7 
is converging, and its sum is the expression for J Xdx. The 
OS es 


fixed limit 6 may be replaced by the variable x, provided no 
values of x are admitted which fail to render series (1) con- 
verging. We should thus have | 


f° Xe = [i usde + [ude + res + [unde | (3). 


232. Formula 3 of the last article still holds true for « = 8, 
even though the series uw, + u, + u,-+ ---, whichis supposed 
converging for « < b, becomes diverging for « =), if, at the 
same time, Series 2 is converging. 


For, however small the quantity @ may be, we have 


fo xax lias u, dex lee TA ee ee, + fade, 


The two members of this equation are continuous functions 
of x, and are constantly equal; hence their limits fora=0 
must be equal, and therefore 


{FP Xdx= fo mde+ fi ee a 


INTEGRATION BY SERIES. 387 
If the series 
2 
F(@) =f (0) + 2f'(0) + po S(O) +, 


to which the development of /(x) by Maclaurin’s Formula 
gives rise, is converging, we shall have 


[fl@)dx = 0+ af (0) + £5 10) + yg SO) 45 


and, if it is wished that this integral should begin with «= 0, 
C must be zero; and we then have 


[Pode =f) + FLO + pag OF 
EXAMPLE 1. {cS =l1+ <2). 


By division, or by the Binomial Formula, we have 


eye's 9 8 eet 
— eet..-toe eet 
ae gt ao” a ada 
—r~— SI cs se é 
eas Tt+e- 4 Waa vera 


When 2 is numerically less than 1, positive or negative, the 
series 1 —v+ x?— @’... is converging, and therefore so also 


2 a3 a4 


is the series « — 3 + ant artsy between the same limits for 


>-1 
x: hence, when «<4,? 


ze ge4 
Cee ae~ 2 ee ies 
It may be shown by direct demonstration, that [~ a" dn 


converges towards 0 as n approaches oo. 


° : a4 _ 
For, if x is positive, we have 5 wows as we”: therefore 


x ada ae 
ite eae ae 


388 INTEGRAL CALCULUS. 


n—1 


n+ 1 


in stopping with any term of the series, the error will be less 


Now, as 7 increases, approaches 0; and consequently, 


than the following term, and will be additive or subtractive 
according as the last term taken is of an even or odd order. 


If « is negative, and @ denotes.a number greater than z, 


ig 1 hs 
Te < ee and therefore 


2 w2"dx eiimee 
9 L—aw ~(n+1)\(1— a) 
The limit of the second member of this inequality for n = 00 


but less than 1, we have 


is zero. In this case, the error is always numerically additive. 


When a = I, the series 1— a+ x?— x@*-+---- is no longer 
2 


a3 
converging; but the series x — a a y —--- is (Art. 231), 


and will represent the value of /2: hence 


gnrrl 
We have gS let eh es 
n being an odd number, and positive. Integrating, and taking 
for tan.~'x the least positive arc having w for the tangent, we 


find 


3 5 n n-+1ld. 
pies oo ‘ee 
tan. | @ == & 3 1s $F, [cag 


The series 1 — x? + «*— «a®... ceases to be converging for 


3 5 
2 ==1; but the series x — = -{- " --- is still converging for 
this value of x: hence 
ae te 
tan. oy ae gtk - 


INTEGRATION BY SERIES. 389 


da 
ee in ho 
Ex. 3. ip Vi = = sin 
We have 


1 135 
Vi-a 1+ 5245 ig aa So 


From this, by multiplying by dz, and integrating, we find 
Agr earns on cea (tae aes 
—1 PE, Ne re tae ope 
Sek aa ace iy se a 


: ; on Lee : ; , 
a converging series when 22, ,) since Series 1 is converging 
between these limits. 


The series 
35 
ne 2 a ea ae ice eR we 
Lt 50 +5 gets ae? 


is not converging when «=1; but since, for s=1=sin. 


_ the series 


1 x3 E39. 3° Ls 
OTmipy acs Samo y ka eo ee 


is converging (Art. 231),* we have 


2 
AK 


es 131.1351 
salts stoastaag@t™ 


A still more converging series is found by making 


whence 


* Space does not allow the proof of convergence or divergence when these con- 
ditions are asserted relative to the series involved in the last three examples. (See 
Art. 68.) 


390 INTEGRAL CALCULUS. 


2338. By integrating /(x)dx by parts, we have 
[Ff (2)du = af (x) — fxf’ (x)da, 
fap (ade =" p(x) — f Spr(a)d, 


fe of cde =f" (@) = fda 


The combination of these results gives 
ue a* 
JF (ada = af(x) — tat () =i ta37(*) “fa ee 


ae ee ae fO-Y (a) el i ma 5 Sent (wd 


This is the series of John Bernouilli, and may be advanta- 
geously used in many cases: for example, if /(x) be a rational 
algebraic function of (n —1)™ degree, f™(x) is 0, and the 
series will terminate; or there may be cases when 


can be more readily found than f f(a)dx, or when only an 


approximate value of if “f(«)dx may be required, and the in- 
0 


tegral fe sr@de may be small enough to be neglected 
without sensible error. 
234, Assuming f /(«)de = g(x), and making «=2#-h, 
we have, by Taylor’s Formula, 
(+ h) — 9(%) = he’ (x) + m9" (2) “Era 
But, because f /(x)da = g(a), we have 
J(%) = 9'(%), f'(&) = 9" (x), f" (@) = g” (2). 


INTEGRATION BY SERIES. 391 


These values, substituted in (1), give 


h? h? 
g(a +h) — 9(2) = Bf a) + 5 (2) tog (a) + 
In this series, making «=a, h=6—a, and denoting by 
A,, Ay, A;..., what f(x), f(x), f”(x)..., become under this 
supposition, then g(x +h) — g(a) becomes 
7 
p(0)—g(a)= ff (x)de, 


and we have 
b mr. A, 2 A, preg 3 
J f@de= Ab SUD aa ee (B greet ae) SPs 
This series enables us to find the approximate value of the 
definite integral f fede when 6 — ais sufficiently small to 


make the series converging. When this is not the case, or 
when the series does not converge rapidly enough for our 
purposes, put b— a= na, and take the integral successively 
between the limits a and a+ a,a-+aanda- 2a, and so on, 
denoting the results by 


B. B 
Brat eG a? + as aire 


i 9 C; 3 
Cia+ 19% +753% ret 

D, 2 D, 3 e 
POE ee ENE I ah 


then (Art. 224) we have 


[ Sla)da= (Bit G4 Dy + +-+)0+ (Bat Cot Dat +-)o? 
+ (B3-0;+D,+ +: )a°-+---, 
a series that may be made to converge as rapidly as we please 


by making @ sufficiently small. 


SGT O Nee 
GEOMETRICAL APPLICATIONS. 


QUADRATURE OF PLANE CURVES REFERRED TO RECTILINEAR CO- 
ORDINATES.— QUADRATURE OF PLANE CURVES REFERRED TO 
POLAR CO-ORDINATES. 


235. The quadrature of a curve is the operation 
of finding the area bounded in whole or in part by the curve. 

If w denote the indefinite area limited by the curve, the 
axis of x, and any two ordinates, it was found (Art. 164) that 

du = ydx =f (x)da ; 
y =f (x) being the equation of the curve referred to the rec- 
tangular axis Ox, Oy. 

If it is desired to have the 
area limited on one side by the 
fixed ordinate C'A, correspond- 
ing to the abscissa « = OA =a, 
> the integral must begin at x=a; 


and we have 


U =" f(2)de. 
Finally, if.the area is to be limited on the other side by the 


ordinate BD, corresponding to x = OB =), we have 
u = area ACDB Bale J (x)dax. 
When the co-ordinate axes are oblique, making with each 


other the angle o, then 
u— area ACDB = sin. ie I (au)dz. 


392 


QUADRATURE OF CURVES. 393 


236. The definite integral is the limit of the sum, taken 
between assigned limits, of an infinite number of infinitely 
small areas (Art. 192). Observing that /(x) dx is equivalent 
to f(a) Aa, if we suppose Aw = da to be positive, the element 
J (x) Ax will have the sign of f(x). Consequently the integral 
will represent the difference between the sum of the segments 
situated above the axis of x and the sum of the segments 
situated below. 

If, for example, the ordinate 
changes, as in the figure, from 
positive to negative, and then 
from negative to positive, the 


area between the ordinates AC, 


BD, will be 
f f(@)dx = ACL — LMN+ NBD; 


and if OL =h, ON=k, the sum of these segments will be 


expressed by 
ff sede —f sode+ f so)dn, 


237. If y=/f(x) is the equation of the curve CM, and 
y, = w(x) that of the curve C/I, 
and the area bounded by these 
curves and the ordinates AC, 
BM, corresponding to «=a, 


x2 = b, is required, we have 


Area COMM =f f(a)de —f w(a)de 


=f} 7@)—v@) | an, 


50 


394 . INTEGRAL CALCULUS. ° 


EXAMPLES. 


Examp.eE 1. The family of parabolas is represented by the 


equation y”= px”, m and n being positive. We have 
1 om 
du = ydx = p” x" dz, 


and 


xz lom é 
fiptdhan= pg 
which may be written 
Le ae tie n 
MS en be ee eee 
But xy measures the area of the rec- 
tangle OPMN, contained by the co-ordi- 
nates of the point J Hence, from the 
above formula, we have 
OPM: OPMN::n: m+n, 
OPM: OMN::n 


that is, the arc of the parabola divides the rectangle con- 


structed on the co-ordinates of its extreme point into parts 
having the ratio of n: m. 
Reciprocally, the property just enunciated belo to the 
parabolas alone ; for the proportion 
OPM: OMN:: 0 
may be written 
UsrXY—UIENSM. 
Hence (m+ n) u = ney, and, by differentiation, we have 
(m+ n)du = nady +- nydzx ; 
or, since du = ydza, 
mydx = nady : 
dx dy 


whence m—_—=n 
x x 


QUADRATURE OF CURVES. 395 


Integrating 
nly = mlx 4+- C, or ly® = la™ + C, 
putting lp for C, we have 
ge (pe or. a po™ 
for the general equation of the curves which possess the 
property in question. 
For the ordinary parabola in which n = 2,m=1, we have 


Ex. 2. The hyperbolas referred to their asymptotes are rep- 
resented by the equation x” y” = p, 
m and n being entire and positive 
numbers. 

Assume the asymptotes to be rec- 
tangular, and let WCW be the branch 
of the curve situated in the angle 


xOy. 
Suppose n> m, and let w= area AOMP, OA =a, OP=2; 
then 


x zr 1 _m 
w= f yd = | nn” n da 
ni pie Fide 


n LOY! Adee! aoe 
or a Bee eee hs, 3 | 
ak 


As x increases, so also does wu, or the area ACMP; and x and 


u become infinite at the same time. If, however, we suppose 
PM to be fixed, and a to decrease, the surface, while continu- 
ally increasing, will remain finite ; and at the limit, when a= 0, 


1 n—m 
it reduces to ee pn a ". Hence the surface PUNLZ 
n—-m 


approaches a fixed limit as the point NV approaches the 
asymptote Oy. 


896 INTEGRAL CALCULUS. 


This limit, which may be written xy, bears to the 


m— mM 
rectangle PMBO the constant ratio of n to n—m: since, 
denoting this limit by u, we have 


nN 


A—MIni: cy, Uu= 
n—m 


LY. 

The converse of this is also true; that is, no curves, ex- 

cept those represented by the equation «” y” = p, possess 

this property: for, from the preceding proportion, we have 
u(n —m) = nxy, which, differentiated, gives 
(n — m) du = nady + nydz ; 


from which, by substitution and reduction, we have 


Integrating . ed40 
nly = — mlx ; 


making C= lp, then ly” = I= : hence 2p? ae 
When m=n, the general equation takes the form zy = p, 
which is that of the equilateral hyperbola of the second 


degree ; and we have 


da 
Yer a ydx = p Bie 
and therefore u=ple+ C= pl= 


by making C =a When p = 1 anda=1,we have u=—Iz; 
and the area is then equal to the Napierian logarithm of the 
abscissa. 
Ex. 3. The equation of the circle, referred to its centre and 
rectangular axes, is 
oi 9? eee, of — A/G ee 


QUADRATURE OF CURVES. 397 


and ydxc = */a*— «dx is the differential of the area of a 
segment limited by the axis of y and 
an arbitrary ordinate PM. Denoting 
this area by wu, we have 


af / a — x de. 
Hence (Ex. 2, p. 326) 


1 Se le x 
Us 2 2 YE a eed 
u=zaVa — 2 + 9 sin.~! 7 

From this we deduce the area of the sector OBM; for 


the area of the triangle OMP is measured by 5m ¥/ a? — x, 


which, subtracted from the expression for w, gives 
2 

sector OBM = ee sin,—! za prey} = sin.—! = — ie 

2 a 2 Gen 


that is, the area of a circular sector is measured by its arc 


arc MB: 


multiplied by one-half of the radius. 
Ex. 4. If a and 6 denote the semi-axes of an ellipse, the 
equation of the curve referred to its centre and axes is 


Gey 0m ater. fs 


Let w denote the area of a seg- 
ment bounded by the axis of y and 
any ordinate, as PI; then 


b ar 
w= | / a? — xv? de. 
ad 0 
Describe a circle on 2a as a diame- 


ter, and denote by w/ the area of the 
segment BMPO; then 


My 5) 
wf VJ/ a? — xv? da. 


398 INTEGRAL CALCULUS. 


? b ub 

ows Me Ue sad OTe ee 
a we 
That is, the segment of the ellipse is, to the segment of the 
circle which corresponds to the same abscissa, in the constant 
ratio of 6 to a; and therefore, denoting the entire area of the 
ellipse by A, and that of the circle by 4’, we have 
Ais ee Dee 


and, since 4’ = za’, it follows that 
b 
A=-—2a* = zab. 
a 


Hence the area of the ellipse is a mean proportional be- 
tween the areas of two circles, having for diameters, the one 
the transverse, and the other the conjugate, axis of the ellipse. 

The ordinates PM, PN, are to each other as a to b, and 
hence the triangles OPM, OPN, are in the same proportion ; 
that is, 


OPN tan Neem 2 em 
OPM PM =a)" ut wa 


u—OPN_b | OON_d. 
w — OPM a’ ° OBM™ a’ 


and thus the area of the elliptical sector may be found in 


g 


terms of the area of the corresponding circular sector. 

An ellipse may be divided into any number of equal sectors 
when we know how to effect this division in a circle. It 
would only be necessary to describe a circle on the major 
axis of the ellipse as a diameter, then divide the circle into 
the required number of equal sectors, and through the points 
in which the circumference is divided draw ordinates to the 
major axis of the ellipse. The sectors formed by joining the 
centre with the points in which these ordinates cut the ellipse 
will be equal. 


QUADRATURE OF CURVES. 399 


Ex. 5. The equation of hyperbola is a? y? — b? x? = — a’b’, 


or y=" Va? a2; and the area of J 


M 
the segment 4//P is expressed by 
Oh ieee 
w=— f Va — a de. 0} A P x 


Hence (Ex. 6, p. 328) 


AMP = 90 Vm a —Fi(2t VEO), 


2 a 
Ex. 6. The differential equation of the cycloid (Art. 146) is 
dy 
Tete aides ee, 
4 Nos; : / 2ry — y? 
a Oe Pea ay 
. U = fyte= J Fo 


This integral may be found by Ex. 1, p. 370: the following, 
however, is a more simple process. 
Put NM = 2r —y =z; then, denoting the area OLNM by 
wu’, we have 
ul = fada = [(2r —y) da = f/2ry — dy ; 
observing that the limits between which these integrals are 
taken must correspond to z= 2r and z=>2r—y. But 


[V2 —¥ dy is evidently the expression for the area of the 
segment of a circle of which 
ris the radius; the segment 
taking its origin at the ex- 
tremity of a diameter, and 
having y for its base. This 


segment is represented by 
ADB. The area OL NM takes its origin from OZ, and the cir- 


cular segment from the point A, and both areas are zero when 


400 INTEGRAL CALCULUS. 


y =0: hence the constant of integration is 0,and we constantly 
have area OLNM= segment ADB. 
When y = 2r, the segment becomes the semicircle ADC, 


ig ee 1 
which is measured by a But 


rectangle OLCA = OA X AC =ar X 2r = 277"; 


that is, the rectangle is twice the area of the generating cir- 
cle: hence 

g 
2 


and therefore the area bounded by a single branch of the 


area semi-cycloid OMCA = 


2 


cycloid and its base is three times the area of the generating 
circle; or, in other words, this area is three-fourths of the rec- 
tangle having for its base the circumference, and its altitude 
the diameter, of the generating circle. 

Although the area of the cycloid may be said to be thus 
represented by a part of a rectangle, it is not a quadrable 
curve; for the base of the rectangle cannot be accurately 
determined by geometrical processes. 

238. Quadrature of curves referred to polar co-ordinates. 
The differential of the area 


PRM (Art. 165) is du = ; rdo: 
hence 
S U == f rray, 


the limits of this integral being 


g the values of 6 corresponding to 
the points & and I. 

Example 1. Applying this formula to the logarithmic spiral, 
of which the equation is 7 = ae”?, we have 


QUADRATURE OF CURVES. AQ] 


one a* ; 2m@g dé — a? 2meg neat ad 
wat fe et o=_ te 
Put PI =,', and in the formula 
make r= r’; then R 
yl? y!? 


A —————_———. 
1 ike 
eee ao ae ") ay, 


The figure supposes PA to be the initial position of the 
radius vector; that is, the position at which 6 = 0 and 
r = PA =a, and also that 6 is positive when the motion of the 
radius vector is in the direction of the motion of the hands of 
a watch. Hence, when the generating point moves in the direc- 
tion from A towards B, 0 is negative. Let the motion take 
place in this direction from the fixed radius vector PR = 7; 
then, after an infinite number of revolutions, 7’ becomes 0, 

yr? 


and the expression for wu reduces to u = i 
m 


2. When the length of the radius vector of a spiral is pro- 
portional to the angle through which it has moved from its 
initial position, its extremity describes the spiral of Archi- 
medes. The equation of this ‘spiral is r= a6; and hence 
ry = a when 6 = 57°.29578 of the circumference of a circle to 
the radius 1. 


For this spiral, we have 
ee 2 pil 292 peel 293 : 
w= 5) rido = 5 | a20%do= a9 + 0; 


and, if the area begins when 6=0, 


C= 0, and u = = ag? When 6 = 2z, 


u — area PAB a a’ is the area de- 


scribed by the radius vector during the first revolution. In 
51 


402 INTEGRAL CALCULUS. 


the second revolution, the radius vector again describes this 
area, and also the area PBA'B' included between the first and 
second spires. Hence the area PBA’B' is measured by 


ey giag 3 __ 28 2 93 
ri (477) (2x)?! = a mt. 


It is evident that during any, as the m, revolution, the 
radius vector describes the whole area out to the m™ spire, 
and that, to find this area, the integral 


eee: 242 ra: 293 
waz farords == a0 


must be taken between the limits (m — 1)2a and 2ma, which 
will give for this area denoted by w” 


alae a*(m2n)* — ; a*(m — 1)? (2)? 


~ 702(2n)?| m® — (m —1)3I. 


In like manner, we have for the entire area denoted by w’, 
out to the (m— 1)" spire, 


w' == a? (2n)? (m — 1)? —(m — 2)*t: 


“ui —4' = 7 a3(2n)?| m* — 2(m— 1)? + (m — 2)%, 


which is the expression for the area included between the 
(m — 1)™ and the m"™ spires. 


If we suppose a = , this formula becomes 
wu" — 14! = % Oat | m?— 2 (m— 1)8-+ (m —2)! 
2 ae AS. 3 am 3 

maddie aaa cad Tu bik 4) at ee 


and in this, making m= 2, we find 2m for the area included 
between the 1* and 2° spires. Hence the area included be- 


QUADRATURE OF CURVES. 403 


tween the (m—1)™ and m™ spires is m— 1 times that 
included between the 1*t and 2° spires. 

239. The quadrature of curvilinear areas is sometimes 
facilitated by transforming rectilinear into polar co-ordinates. 

Take, for example, the foliwm of Descartes, which, referred 
to rectangular axes, is represent- 
ed by the equation 
v* + y* — axy = 0. 

This curve is composed of two 
branches, infinite in extent, which 
intersect at the origin of co-ordi- 


nates, and which have for a com- 
mon asymptote the straight line of which the equation is 


a 


To determine the area of any portion of this curve in terms 
of the primitive co-ordinates, we must find what the integral of 
ydx becomes when in it the value of y derived from the equa- 
tion of the curve is substituted. This requires the solution of 
an equation of the third degree ; but if rectilinear be changed 
into polar co-ordinates, the pole being at O, there will be but 
one value of the radius vector in any assumed direction; for, 
the origin being a double point, two values of 7, each equal to 
zero, must satisfy the polar equation of the curve, and the 
first member of this equation must be divisible by 7”. 

Ox being the polar axis, the transformed equation is 


r*(cos.? 6 + sin.? 0) — ay* sin. 0 cos. 6 = 0: 


whence __ asin. 6 cos. 6 
~ gin. é + cos. 6 
6 


1 , 
For the area of the segment OMN, we have u = a J ro, 


404 INTEGRAL CALCULUS. 


which, by substituting for 7 its value above, becomes 


a ke ? sin. cos.? 4 Dak pe sin.” 6d0 
2 » (cos.20-+-sin.29)? ~~ 2/ , cos.40(1 + tan36)? 
6 
i] 
2 
a? ee cos. 6 


"21 (tan 
0 


To effect this integration, put 


1-+ tant O0=2: .°. dz=3 tan2¢6 — 
and hence 
Sane ee 
cos."6 1 pdz fal 
(isttanZ¢}e7 8 Pees 1! 
i; 
Bevo RE TAT) 
a? 1 
. a ee 
o*. uf 6 Taotanee Y 


a? 
The area beginning when 9 = 0, we have C==, and con- 
sequently 
fr a? tan.* 0 
7a) 6 det tans 


The entire area OMDZ is found by making 6 = 5 in the 


2 
“valuc of u, which then becomes alg for then the fraction 


6 
TaD. 0) hae 
“Aertan 0° 


SECTION VI. 


RECTIFICATION OF PLANE CURVES. 


240. Tue rectification of a curve is the operation of find- 
ing its length, and the curve is said to be rectufiable when this 
length can be represented by a straight line. 

Denoting by s the arc of a curve comprised between a fixed 
point and an arbitrary point (x, y), we have 


Se eee Se 2 
ds = / daz? + dy? = de) + we (Art. 161); 
and, by integration, 
a i 
cael bE ca ATE 
: J x | eels da? 
By means of the equation of the curve, ds may be expressed 
in terms of either x or y; and, the integral being then taken 


between the assigned limits, we have the length of the curve. 
ExampPLe 1. The Common Parabola. From the equa- 


tion y? = 2px of the curve, we find ydy = pdx, dx = ee. 
This value of dz, substituted in the differential formula, gives 
Tne Sie Ch ee 
da [UU + dy = Vy Ep 
on a ae 
whence, making the are begin at the vertex of the parabola, 


1 py 1 Ee 
s= = da ata? 
uk yvy p 


= VP te t+ gly t Vy FP) + C, 


by Ex. 5, page 327. 
405 


406 INTEGRAL CALCULUS. 


Since the integral is to be zero, for y = 0 we have 


jes Bey Recon IE 
O= Ge Te: o= 5 Ps 
By the substitution of this value of C, the formula becomes 


ep pane Mawes yt vie 


Ex. 2. The Ellipse. From the equation of the curve 
ay? + b2 a? — 7b 


we get 

BY Wenn ae 
dx ary 
hence 


ph jgel | pi kbtet Pipe 
ds = de | pee =e fi ae a?(a2b? — b?)’ 


PY ane PPC ER) 1 PMS ES 
1 — 10 oe a? —e*a 
or ds =dz.| ( ) tc ae 


a? (a? — &*) ea 


in which e= 


vie is the eccentricity of the ellipse. 

Suppose the arc CN to be estimated from the vertex C 
of the minor axis; then, to get the 
length of the arc CNA, the integral 
of the expression for ds must be 
taken between the limits a = 0 and 
ea; but all the values of & 
between 0 and a will be given by 


2 —asin.g, the angle g varying be- 


tween 0 and 5" The substitution in 


the value of ds of these values of x and its differential gives 


ds=av/1 — e’sin’ gd; 


RECTIFICATION OF CURVES.  (AOT 


and therefore 


pF afl — e*sin’ g dg. 
0 


This integral belongs to a class of functions for which we have 
no expression except under the sign of integration ; and, to find 
its approximate value, we must have recourse to a series. 
The Binomial Formula gives 


nie $ ae ae ee oe 
(1 — e?sin2g)° = 1 — ge sing a5 qe sin.’ 


Lg Ss 
Sow a 


hence, for the arc C/N, we have 


€ aeons 


e®sin.®g — ne 
6 


let 
24 


CO] Nr 


sag — : ae? f sin? gd — 5 sy qe f sins gd 


moa Jf sin. gdp... 


The integrals in the second member of this value of s may 
be found by applying Formula 1 of Art. 221. We should thus 


get, by taking all the Sa Nee between the limits 0 and 5 


a1), 


in 24/1 — e?sin. *gdg = 5 — 


es 
A 
246 
hence, for the arc CNA, we have 


rh LNT a EN ~ 1 (1-3 5oN 
2 kak Se ea Pere ey Of Yes 
age ae (*) AG ) ere) 


SEGRE). 
aaa ‘) 


This is a converging series, and the more rapidly so as e 
becomes less, or as @ and 6 approach equality. When the 


408 INTEGRAL CALCULUS. 


eccentricity is very small, it would be sufficient to compute 
but a few of the terms of the series. 
This value of the arc CNA may be found without using 


Formula 1 of Art. 221; for, assuming the first equation in that 
nm 


6)? 
2 


article, and taking the integral between the limits 0 and 


we have 
Tv 


7: ie a 
(Bs sin.” gdg = ce le sin.” —* pd. 
; m 


0 


In like manner, 


a er bie a 
fi sinetgdy =P 


T 


= ne 0 he 
2 ain m—4 — 2 ain m—6 
iE sit. a dg = ; ai) sin. aS, 


Multiplymg these equations member by member, there 
results 
— a __ (m—1)(m — 3)(m —5)...8 mt 
ip spiel fe las m(m — 2)(m—4)...4.2 2 
The values given by this, by making m equal to 2, 4, 6,..., 


successively substituted in the value of s, lead to the result 
before found. 

The angle g is found by the following construction: On the 
major axis as a diameter describe a circle; produce the ordi- 
nate PN to meet the circumference at J, and draw OM; then 

e= OP OM cos. POM = asmib eae 
hence mg = angle BOM. 


Kx. 3. The Hyperbola. Assuming the equation 


ary? 2? — oq) 


RECTIFICATION OF CURVES. 409 


of this curve, and proceeding as in the case of the ellipse, we 
get 


i San (et b?)x:? ie at 


a? (x? es a”) 


To simplify, put a? + 6? = ae; then 

Atenas v2 

ds =da J iba saaed . 
x —a 
Now, for one branch of the hyperbola, all admissible values 

of « are comprised between + a and +o, and for the other 
branch such values are comprised between — a and — 0; and 
it is evident that all of these values will be given by the 


: a . nm 
tio ch a ad @) kine vary between 0 and — f 
equation x ee y making 9 y n 5 or 


one branch, and between : and z for the other. 


ad 


Substituting this value of x, we have 


Di asin. pd 
cos2g ’ 
a*e* — a*cos.? ae Cos.” 
2p cos.” ~ 


whence (fig. Ex. 5, p. 899) 
cer [Ge 1 ij aaa 
0 


COS.” @ e 


Developing the radical in this integral, we get 


sae e sts Cosy qe rai COS. 
y COS.” p 


Dees 24 e' 
1.1.3.5...(2% — 3) ore 
ha YP; 


2.4,.6...27 en 
or $8 = ae tan mag 
4g Sr ihoaeey 
@ pel 1 cos’g , 11 3 cos. 
ahs 2 e? 5 2 G e* 13 oe 


52 


410 INTEGRAL CALCULUS. 


The integration now depends on that of expressions of the 
form cos.” gdg, and may be effected by the application of For- 


yh 


mula 1, Art. 221, after changing in it x into 5 


Ex. 4. The Cycloid. The differential equation of this 
curve (Art. 146) is 


Pare yp: ¢ Vfl); 
dan = dy || J yor dy = de 
2r—y y 


In the formula ds = da? + dy?, replacing dx by its value, 
we get 


ds = V 2r(2r — y) dy: 
s= — 2W 2r(24r—y)+-C. 


If s be estimated from the origin O to the right, we must 


have 

0— —4r+C: .*. C=4r, 
and 
s= OP = 4r— 20/ 2r(2r —y). 

In this, making y = 2r, we 
have OO’, the semiare of the 
cycloid, equal to 47, and the 
whole arc therefore equal to 
87, or four times the diameter of the generating circle. 

Estimating the arc from the vertex O/ to the left, then 
C= 0, since at this point y = 2r; and we have 

OPS 2r/2r(Qr — y). 

But / 2r(Qr — yy) =V GNX GO=PGE: 
hence arc O’P =2 chord PG; that is, the length of the arc 
of a cycloid, estimated from the vertex, is twice the corre- 
sponding chord of the generating circle. 


L F &£ 


SECTION VIL. 


DOUBLE INTEGRATION. — TRIPLE INTEGRATION. 


241. Double Integrals are expressions involving two 
integrals with respect to different variables. Suppose it is 
required to find the value of w which will satisfy the equation 


; ? 
tas = (a, y), the variables x and y being independent. 


This equation may be written 


ad du =*, 


by making v = a - The function v must be such, that its 
differential co-efficient with respect to y, x being considered 
as constant, is equal g(x,y). We therefore have 
du 
eR =" 
hence w must be such a function of « and y that its differen- 
tial co-efficient with respect to x, y being constant, is equal to 


fo (x, y)dy; and therefore 
w=} fol y)dy | de. 


The value of uw is thus obtained by integrating the original 
expression with respect to y, and then integrating the result 
with respect to a. 

The last equation is generally and more concisely written 

py | 92, y)dady, or u= ff p(x, y)dydz ; 


411 


412 INTEGRAL CALCULUS. 


the first form indicating that the first integration is performed 
with respect to x, and the second integration with respect to 
y. The second form indicates that the order of integration is 
reversed. 

du _ du 
didy  dydx 


these partial differential co-efficients were the same, in which- 


242. It was shown (Art. 91) that 


, or that 


ever order, with respect to # and y, the differentiation is per- 
formed. We will now prove that the result of the integration 
in the one order can differ from that obtained in the other 
only by the sum of two arbitrary functions, the one of a, and 
the other of y. Let u,, w., be two functions of x and y, either 


of which satisfies the equation iy g(x,y); then 


d?u 
dee = 9(#,Y), Tage 9% y): 
: au, Ay 0 
ie dxdy  dady~ 
d /dv : 
or =f a ante — . 
ps @ 0, putting v =u, — uy, 


Now, 


dv 
dy cannot be a function o. x, otherwise its differential 


co-efficient with respect to # could not be 0; but it may be 


any function of y. Hence we may put 


a= = f(y); whence v= ffay + x(x), 


in which ie denotes an arbitrary function of # Putting 


fiyyy = w(y), w(y) being as arbitrary as f(y), we have 


finally 
Y= U—M=Y(y) +7(2), 
as it was proposed to prove. 


DOUBLE INTEGRATION. 413 


b 
243. A double integral J if acy vdety is the HAW Oe 
avg 


all the products of the form g(a, y)AxAy between the limits 
of integration. Let (a, y) be a function of & and y, which 
remains finite and continuous for values of x between a and b, 
and for values of y between « and 6. 

To abbreviate, put g(x,y) =z. Now, if we suppose x to 
be constant while y varies between the limits « and £, we 
have (Art. 192) 


ff zdy = lim. S2ay. 
a 


Multiplying both members of this equation by Aa, and sup- 
posing # to vary between the limits a and 6 while y remains 
constant, there results 


zac f" zdy = Sax lim. Seay : 
hence lim. zac f zdy =lim.£Axlim.Szay=lim. SSzaxay. 
ot 


G oa 
But lim. Sax J 2zdy = zdacd. 
JJ, J. edeay 
by the article above referred to: therefore 
Pere 
ie if g(x, y)dady = lim. SS q(x, y)Axay. 

Writers do not agree as to the notation for double integrals ; 
some making the first sign f refer to the variable whose dif- 
ferential comes first in the integral, while others make the 
first sign fe refer to the other variable. In what follows, the 


first sign Bk: will relate to the variable whose differential is first 


written in the indicated integral. 


A414 INTEGRAL CALCULUS. 


244. In the last article, it was supposed that the variables 
x and y were independent. It is sometimes the case, how- 
ever, that the limits in the first integration are functions of 


b 8 
the other variable. For example, let a} i) p(x, y)dady be 
a a 


the required integral in which « = 7(a), and 6 =w(«); then 


b b (x) 
ip ips g(x, y)dady = if f9@ y) dady. 
Suppose J’(x, y) to be the result obtained by integrating, 
first with respect to y, regarding # as constant; then, for the 
integral between the assigned limits for y, we have 


F \ 2, »(2)| =F fa, ye) t; 
and finally 


> py(z) : 
Nile g(x, y)dady = tp (F {e, w(x) —Ff |e, 1(2)} )de. 

When the limits of a double integral are constant, it is im- 
material in what order, with respect to the variables, the 
integration is effected; that is, a change in the order of in- 
tegration does not require a change in the values of the limits. 
But when the limits for one variable are functions of the other 
variable, and the order of integration is changed, a special 
investigation is necessary to determine what the new limits 
must be to preserve the equality of the results. A geometri- 
cal illustration of this will be given in the next section. 

245. Triple Integration. Let it be required to de- 
termine a function w of the three independent variables a, y, z, 


C Cie : 
which will satisfy the equation adv V. We may write 
a ds dhe. 
dadyda — da dady ~"? 

d*u d d*u 


or 


acai dz =F de dz—= Vdz: 


TRIPLE INTEGRATION. — A15 


hence by integration with respect to z, regarding w and y as 
constant, 
du ‘ 


T” being an arbitrary function of z and y. Again: we have 


du d du 1 
eae d du 


or 


ratty, 2 mer att = dy f Vda+ Th ay, 


which, by integrating with respect to y, « and z being con- 
stant, gives 


ot — fay f Vide + T+ 8"; 


T’ being an arbitrary function of x and y coming from f 7" dy, 
and §’ an arbitrary function of x and 2. 
Finally 


d 
ux f 7, du= fdefdyfVde+ T+ S+R; 


T, Rk, and S being arbitrary functions, — the first of w and y 
resulting from Je T' dx, the second of x and z resulting from 
fS'da, and the third of y and z. 


It is usual to write the differentials together after the last 


sign of integration: the above equation thus becomes 


u=f ff Vdadyde+ T+ S+ &. 


This example suffices to show the manner of passing from 
a differential co-efficient of any order of a function of several 
variables back to the function itself. When the variables are 
independent of each other, as has been here supposed, there 


416 INTEGRAL CALCULUS. 


is no dependence between the arbitrary functions 7, 8, 2; 
but more commonly at the limits of the integral the variables 
are not independent of each other. For example, the limits of 
the integral with respect to z may correspond to z = Ex, Y), 
z= f(x,y); those with respect to y, to y=/(x), y=f, (a); 
and, finally, those with respect to x, tox=a,x=—). 

By a demonstration similar to that given in the case of a 
double integral (Art. 244), it may be shown that 


BOM PRE; ue here 
1s dx J dy p(x, y, 2)dz= lim. SSZArAYy AZ. 


SECTION VIII. 


QUADRATURE OF CURVED SURFACES. — CUBATURE OF SOLIDS. 


246. Let F(x, y, z)=0 be the equation of any surface 
whatever, and take on this surface the point P, (x, y, 2), and 
the adjacent point Q, («+ au, y+ ay,2+ Az). Project these 
points in P’, Q’, on the plane 
x, y, and construct the rec- 
tangle P’Q’ by drawing par- 
allels to the axes Ox, Oy. The 
lateral faces of the right prism 
of which P’Q’ is the base will 
intercept the element PQ of 
the curved surface. Denote 
by 2. the angle that the tangent 
plane to the surface at the 


point P makes with the plane 
(x,y). This plane is determined by the tangent lines drawn 
to the curves Pq, Pp, at the point P. The tangent line to the 
dz 
dic 


is the tangent, and the tangent line to the second makes with 


first curve makes with the axis of 2 an angle of which 


dz 
dy 


are the angles which the traces of the plane of these two lines, 


the axis of y an angle of which is the tangent. These 


that is, of the tangent plane to the surface at the point P on 


the planes (z, x), (z,¥), make with the same axes. Now, from_ 
58 417 


418 INTEGRAL CALCULUS. 


propositions 1 and 3, chap. ix., Robinson’s “ Analytical Geome- 
try,” we readily find, without regard to sign, 


Eas A 
H+) +@) 


The rectangle P’Q’ is measured by Away, and is the pro- 


COs: A= 


jection on the plane (x, y) of the corresponding element of the 


AwA 
Z, hence, 
cos.2 


tangent plane. This element is measured by 


for the element of the tangent plane, we have 


oat dz\? dz\? A ° 
cos;A +(F) is ae meh! 


= sec.AAvAay. 


Let S denote any extent of the surface under considera- 
tion, and assume that the limit of the sum of the terms 
sec. AAxvAy, for all values of x and y between assigned limits, 
is the area of the surface; then 


B= ff 1+ e) Ha) 5 am 


If the surface is limited by two planes parallel to the plane 
(z,y) at the distances « =a,x=b, and by the surfaces of 
two right cylinders whose bases are represented by the equa- 
tions y = g(x), y= w(x), we should have 


sofia ffir (EY «(Ne 


and, when the cylindrical surfaces reduce to planes parallel to 
the plane (zx), p() and w(x) become constants c and e, and 
the formula reduces to 


cafes (2) (ita 


VOLUMES OF SOLIDS. 419 


247. Area of Surfaces of Revolution. If y=/(«x) 
be the equation of a curve referred to rectangular axes, the 
differential co-efficient of the area of the surface generated 
by the revolution of this curve about the axis of a has been 
_ found (Art. 167) to be 


ds _ i aN 
ae tees ede Nl 


23 ius an f J1+ (32) ya 


248. Volumes of Solids. Consider the volume bounded 
by the surface of which F(a, y,z)=0 is the equation; and 


through the point P, (x, y, z), in this surface, pass planes par- 
allel to the planes (2, x), (z, y); and also through the point 
Q, (w+ Ax, y+ Ay, a+ Az), 
adjacent to the point P, pass 
planes parallel to the same 
co-ordinate planes. These 
four planes are the lateral 
boundaries of a prismatic col- 
umn, having P’Q for its base, 
and terminated above by the 
element PQ of the curved 
surface. The volume of this 


column is measured by zaxay, 

when Az, Ay, are decreased without limit; and the volume 
_ bounded by any portion of the curved surface, the plane 
(x, y) and planes parallel to the planes (2, y), (z, x), will be 
the limit of the sum of a series of terms of which zAwvay 1s 
the type. Denoting this volume by V, we have 


V=Zzavay = f fedudy. 


42.0 INTEGRAL CALCULUS. 


From the equation F(x, y, 2) = 0, which is the equation of 
the surface, we have z= g(a, y). If we integrate first with 
respect to y, we get the sum of the columns forming a layer, 
included between two planes perpendicular to the axis of a; 
and hence the limits of integration with respect to y become 
functions of x, and we should have fady =f(x); f(x) being, 


in fact, the area of the section of the solid made by a plane 
parallel to the plane (z, y). Thence, finally, V= if J (x) dz. 


249. Volumes of Solids of Revolution. The differ- 
ential co-efficient of the volume generated by the revolution, 
about the axis of x, of the plane area bounded on the one 
side by the axis of ~, and on the other by the curve having 
y =/f(«x) for its equation, has been found (Art. 166) to be 


or my” = f(x): 
hence, by integration, 
Vn fyde Ti aoe 
Here, as was the case at the end of the last article, f(x) = my? 
is the area of a section of the solid made by a plane perpen- 
dicular to the axis of x; and the integral is the expression for 


the sum of the elementary slices into which we may conceive 
the solid to be divided by such planes. 


APPLICATIONS. 


Wu EXAMPLE 1. Required the measure 

M’ of the zone generated by the revolu- 

tion of the arc MM' of a circle about 

B OPPA X_ thediameter BA. The equation of the 

circle is 2? + y= R*. Denoting the area of the zone by S, 
if OP =a, OP’ =}, we shall have (Art. 247) 


EXAMPLES. 
0 dy\2 
S= 2n f y Jit (fh) de 
eae on | Ra 
=2af iy ON, w= On fs x 


eel (ba) — Ink x PP’ 


421 


To get the entire surface of the sphere, the integral must be 


taken between the limits x= — Rk, x= Rh, which will give 
S=4a2h’. 

Ex. 2. Suppose the ellipse of which 
the arc BMA is a quadrant to revolve B M 


about its transverse axis: required the eee 
measure of the surface generated by 


the portion BM of this arc, begining 


at the extremity of the conjugate axis. We now have 


gain fy [1+ | +(3 i 


From the equation of the ellipse, a’y? + b’x’? = a*b’, we 


2 
get - = — a whence 


dx 


yit(2) = Vay ia _ bat (GB ye , 


ay ay 


and finally, by making */a? — b? = ae, we have 


| dy\? _ b/a? — ea? 
J td ary if ay 


therefore 


i atic Seana - Ibe 
Sane | Jai — eat de = sie oe a? de. 


422 INTEGRAL CALCULUS. 


But (Ex. 2, p. 326) 
z le? x er 7 |g" la? 
fo Qo atte aba Eat 5 Soin Se 


therefore 
abe a a? ex 
S= —(a«.J/— —2’? +5 sin- 7 Se 
2 
a e a 


If, in this expression, we make «=a, and take twice the 
result, we get 


S = 2ab? +. —— ae sin.—le 


for the entire surface of the ane ellipsoid of revolution. 


Suppose, now, that a < b, or that the ellipse is revolved 
about its conjugate axis, and put «/b?—a?—be; then we 
shall have 

Gacrlae br/at Oe ee ae 
0 ary 
But (Ex. ‘ p. His 


ss Be 1 rs iN: a . 
seade os Prt ns pos) rat ashe ee 13 
Sige te ae = TA peta + apa le+ aie 


therefore 


mb" e as P at | @ 
na ee i ° aaa ea Chas hte pate) +e 


Since this integral should be zero, for « = 0 we have 


syed 


hence 


CUBATURE OF SOLIDS. 433 


If in this we make x = a, and take twice the result, we shall 


have 


Bean a2 Bie’ qe cso aria , 


for the entire surface of the oblate ellipsoid of revolution. 
If we suppose a = b, and therefore e =0, the second term 


0 
in the last expression for S takes the form’ 4 ; but, by the rule 


for the evaluation of indeterminate forms, we readily find 


ge et.) 
afk 


a 
€ 


lim. 


whence we have 47a? for the surface of the sphere. 


Ex. 3. Cubature of the Ellipsoid of Revolution. 
The equation of the ellipse, referred to its major axis and 


9 


a 


the tangent line at its vertex, is y? = — (2ax — x”); and there- 
a“ 


fore, for the volume of the ellipsoid, we have (Art. 249) 
ab? mb* ( . =) 
se : ae Ax — aaa 
a” 3 


To get the entire volume, we make x = 2a; and then 


This is the volume of the prolate ellipsoid. To get that of 
the oblate ellipsoid, a and 6 must be interchanged in the last 
formula. We thus get, for the measure of the entire volume, 


4 ere ie: f 
gta; from which it is seen that this volume is greater than 


the first. Making a= J, the ellipsoid becomes a sphere, the 


424 INTEGRAL CALCULUS. 


ei 4 : 
volume of which is expressed by 3 eau ; and, for the volume 


of a spherical segment of a single base, the expression is 
7 
3 x*(3a— x). 


Ex. 4. Volume generated by the Revolution of a 
Cycloid about its Base. 


In the formula V= fny?dx, substitute for dx its value 
ydy 
(2ry — y?)* 


and we have 


ies ; derived from the equation of the cycloid. 


JULY EO 
? 


a) 


but (Ex. 1, page 370) 


(2ry —y?) 


3 2 st 2 
fp =-f env) 42 
(Q2ry —y*)° 3 3 ¢ (2h 
lis yidy = — “en y +27 f yy 7? 
(2ry — yy’) . 2) (2ry ae 
f- ydy == (ary — 9) + rf dy : 
2ry — y”)* (27y — y’) 


= — (2ry — y?) +r ver.sin.—! Zs 
ii 


therefore, by substitution and reduction, 


3 
eH, MSE. — a(2ry— wie a “ry +> 12) 
(2ry — y?)" 


as 3 artver. sin.1? +. 


: EXAMPLES. 426 


Taking this integral between the limits y= 0, y = 2r, and 
doubling the result, we have, for the entire volume generated 
by the revolution of a single branch of the cycloid, 

eae one? 

Ex. 5. Volume of an Ellipsoid. Take for the co- 
ordinate axes the principal awes of the ellipsoid. The equation 

; et ge Be oe 
of its surface is then at -- 52 + a 1% 

The section PMM' of the ellipsoid made by a plane parallel 
to the plane ZOy, and at the dis- 
tance OP = x from the origin, has 
for its equation 

2 2 2 
The semi-axes of this section will 


be found by making in succession 


Reel ed 0; they are 


Hier eees & ae 3 
fe ops ot 
a” a” 


bence the area of the section is 


abe 


we 
abe (1 = S Se (a? — x”); 
and, for the volume of the segment included between the 
planes ZOy and PMM’, we have 


be 23 
v= (a a — 2?)dx =~ (@ ato 5): 
To get the volume of half the ellipsoid, make in this formula 


; 2 : 
%—@, which gives V = 5 mabe; and hence the entire volume 


is measured by 5 mabe 


54 


426 INTEGRAL CALCULUS. 


Ex. 6. The areas of surfaces and volumes. of solids have 
thus far been found by single integration. As an example of 
double integration, let it be required to find the volume 
bounded by the surface determined by the equation xy = az, 
and by the four planes having for their equations 


C= 8, C= B,, Y=HYi, Y= Y>- 


The expression for this volume is 


y ‘i fe °S dady = me —y')(«, —«') 
=F (1 — ma — 41) (G1 + aa + 1s FOI) 


if 
ane = ©) (Yo — 91) (41 ee ee 


in which 2,,%., 23, 24, are the ordinates of the points in which 
the lateral edges of the volume considered pierce the surface 
Be) eee, 

Ex. 7. To illustrate triple integration geometrically, in the 
figure suppose planes to be 
passed perpendicular to the 
axis of z. Let two of these 
planes be at the distances z 
and z+ Az respectively from 
the origin of co-ordinates, 
cutting from the elementary 
column PQ’ a rectangular 
parellelopipedon ab measured 
by AgvAyAz. This _parallel- 


opipedon may be considered 


as an element of the whole volume V: hence 


V= ff fdadydz. 


EXAMPLES. 42,7 


Required the portion of the volume of the right cylinder 
that is intercepted by the planes z = xwtan.0, z= a tan.0’; the 
equation of the base of the cylinder being x? + y¥? — 2ax = 0- 
Here the limits of the integral are z= tan.0,z—=a tan. 0’, 
y= — V 2an — 2, y= tv 2ax— a, © == 0; 7 = 2a; there- 
fore, denoting the values of y by — y,, +y,, 


2a a tan, 9! 
| ae ae f_.. dadyde 
0 a= hi x tan. @ 
2a p+y, 
=| He (tan. 6’ — tan. 0) «dady 
0 1 


= 2(tan.6’ — tan.0) f en 200 — «dx 
= (tan. 6’ — tan.) za’. 

The base of this cylinder is a circle in the plane (a, y) tan- 
gent to the axis of y at the origin of co-ordinates; and the 
secant planes pass through the 
origin, and are perpendicular 
to the plane (z, x). The re- 
quired volume is therefore the 
portion of the cylinder included 
between the sections OP, OP’. 
It can be seen from this exam- 
ple why, as was observed in 
Art. 244, when there is a rela- 


tion between the variables at 

the limits of an integral, the order of integration cannot be 
changed without at the same time ascertaining if it be not 
necessary to make a corresponding change in the limiting 
values of the variables. In this case, after integrating with 


respect to z, we integrate with respect to y, taking the inte- 


} 4 
gral between the limits y= —(2ax—’)", y=-+ (2ax—2a’)’; 


428 INTEGRAL CALCULUS. 


that is, the integral is considered as bounded by the circum- 
ference of a circle tangent to the axis of y at the origin; but 
by what portion of the circumference is not specified until the 
limiting values of w are assigned. The integral with respect 
to a is then taken from « = 0 to x = 2a, which thus embraces 
the whole circumference. 

But it is obvious, that, if the order of integration with respect 
to « and y be reversed, then, that the integral may embrace 
the whole base of the cylinder, the limits with respect to x 
must be e=a—WVa?—y?, e=a+WVa?—y?; and those 
with respect to y must be y= —a, y= +a. We now have, 


denoting the limiting values of x by x,, — &, 


esti hk Hid alee i dydaxdz 
iy fi (tan.0’ — tan. 6) adydx 


2 2af (tana! — tan.0)V a? — y?dy 


= (tan.6’ — tan.@)ma* (Ex. 2, page 326); 
which agrees with the first result. 

250. Polar Formula. The polar equation of a plane 
curve being r = (9), if s denote the length of an are of the 
curve estimated from a fixed point, the differential co-efficient 
of this are (Art. ae is 


s=frs(Gyp 
fiee(ia on 


or, by taking r as the independent variable, 


ea fin ) it. 1a (2). 


POLAR FORMULAE. - 429 


ExampLE 1. Applying Formula 1 to the spiral of Archime- 
des, the equation of which is r = a4, we have 


s= f(r? + a?)8 dd =af(1 + 0°)? do 
Gf] 1 ‘7 
=F (L+oyrtotia+(+oy +e. 


If the are considered begins at the pole where @ = 0, then 
=). 


Ex. 2. For the logarithmic spiral, we have r= da’, or 


r = bee by making a = ae Hopi page! a and a = O whence, 
by Formula 2, 
= f/(lte)dr=/f/(l+ec)r+C. 
If the limits of the integral correspond to the radii vectores 
7, 11, the length of the arc is 


8 /(1 +’) (7) — 79). 


: dd , : 
Since rv — is the expression for the tangent of the angle 


dr 
made by the radius vector of the curve at any point and the 
tangent line at that point, we have, calling this angle @q, . 
ds 
tan.a@—c; hence sec.a = 4/(1 + c”), and Fp a BOG. a: there- 
r 
fore s=rsec.a-+C,and the definite portion of the arc an- 
swering to 19, 7;, 18 (7; —7o) Sec. &. 
251. To find the length of a curve in terms of the radius 
vector and the perpendicular demitted from the pole to the 


: 
tangent line to the curve at any point, we have cos.¢ = en 
8 


(cor. Art. 163) : hence, if p denotes the length of the perpen- 
dicular, 
p te Pj AUN Tp 


0. © — >» cos. = 
, 


r 4 ak r : 


430 INTEGRAL CALCULUS. 


ia 
therefore 3 


—~ a/ 7? — p? aed byes oe r? — pe : 

252. The ae of a curve may also be expressed in 

terms of the perpendicular and its inclination to the initial 
line. 

Let w and y be the co-ordinates of any point VM of the curve, 
and denote by s the length 
of the curve included be- 
tween the fixed point 4 
and the point MM. From 
the origin Jet fall the per- 
pendicular OP upon the 
tangent to the curve at the 

* point M,and make OP=yp, 
MP =u, and the angle POx=0; then, from the figure, we 
readily find 


p—=x«cos.6 + ysin. 6, 


u = x sin. 0 — y cos. 6. 
Also we have 


dy t. 0 a sec. 0 
— = — cot, 6,—-— =) eoseed. 
dx aa: 
therefore 
d d. 
oie —xsin.@ + ycos.6-+ cos. 0 >t sin. 0 
! oh 
But, since 6 is the independent variable, 7 cot.@ may 
dy 
: do cos. 6 
be written dx = See whence 
dd 


da 
cy Sige 
sin. 0! 5 t 008: py 


POLAR FORMULZ. * 430 


oF in. 0 cos. 0 
= — sin. o0=— 
dé s oe Y U; 
d’*p du : _ ae 9%. 
> x cos.@ —ysin. # — sin. 6 16 -+ cos. fa 
dy 
But, from ae cot. 6, we get 
dy , da 
COS. 0 ap aoa cos.” cosec. 0 at 
a LL re FUN MUP cok eco. A 
- — sin. eet oS: fake pee aac Mee 
dx /sin.?6 +- cos.?6 dc 
eee) COSC. 7 * 
do sin. 0 ) do 
ods ; ds dx 
The equation i ae cosec.@ gives nee cosec. 0 Th hence, 
by substitution, 
oh ae e i ae as 
am cos.d — y sin. LET 
ds 
—=—p + PS 
therefore 
dp 
dp 
° ee 4 
a 8 sat J pa ; 
or st+u= fpdo. 


Taking the integral between the limits 4), 6,, 80) $1) Uo, M15 
being the corresponding values of s and wu, we have 


stm —u = fi pd. 


The sign of w will be positive or Ne. according as the 
angle POx = 6 is greater or less than the angle MOx. These 


432, INTEGRAL CALCULUS. 


results may be used for several purposes, the most important 
of which are, — 

First, To find the length of any portion of a curve, the 
equation of the curve being given. In this case, from the 


arene! 
equation of the curve and the equation oF aa 6,cand y, 


dx 
and therefore p = «cos.@-+ ysin.6, can be determined in 
terms of 0; and, by integration, s may be found from the 


dp 
equation s = aA + [pdo. 


Second, To find a curve, the length of a portion ‘of which 
shall represent a proposed integral. Here, if the integral be 
fpdo, p being a function of 6, the equation of the curve is 


found by eliminating 6 between the equations 


dp. : d 
== p COS, 0 do S10. 0, app sin. 0 + 47 cos. 6, 
hich we get from the equations 
p=xcos.6+ ysin.6, oe = — xsin.6 + y cos. 6. 
The proposed integral will then be represented by s — a0 
APPLICATION. 


Let BUC be a quadrant 
of an ellipse of which the 
equation referred to its 
centre and axes is 

ak fb b2 a? — a” 52, 
This equation, by making 
*= a?(1 — e?), may be 
put under the form 
y? = (1 — e’) (a? — a’). 


POLAR FORMULZ. 433 


Make POx — #, 


Then, from the properties of this curve, we have 
tse 4 | mae Ne 
On? = a; tan. PTO = cot.?6= Coed. 
x — 2 
From the last equation, we get 
ers: eri 62 
sin? =~, dhe ue ae 
a* — e* a Coe Cee? 
aa? aH : a?(1—e? 
“te OP’ = 0T" x costo = a2 St = ©), 
eee} AMD aE A 
or OP =p=a [eG = ayia e ante 
Therefore 


OM+ MP =s+usafV1— eosin? di. 


It is here supposed that the integral takes its origin at C, 
the vertex of the transverse axis. Now, if the point A be so 
taken that the angle BOA = 6, it has been shown (Art. 240) 
that 

Are BA=afV1—e? sin?0do: 


ag OM + MP = BA. 


Also we have 
dp ae*sin.@cos.6 , 
dp /1 —e? sin? 


and, w being the abscissa of the point I, 


EE as 


po eink 


dé 


ae? sin. 6 cos. on a cos. 6@ 


= a(1— e?sin.0)* cos. 6+ eee SS ae 
ia —e’sin. 29)? (1 — e? sin.? 0) 


55 


434 INTEGRAL .CALCULUS. 


Therefore UP = e*xsin.6; and, x’ being the abscissa of A, 
era | 
we have x’=asin.6: .*, MP= ae and hence 


2 
BA — CM = UP =~ ax’, 


a result known as Fagnani’s Theorem. 


From the values of # and x’, we get 


2 
2 a—a’sin2?¢@ a?—ae’ | 


x ae 
1 — e’ sin.2 6 : 


2 
e2a/ 
a” 


which gives 
2 2 
e* a xe! —a*(a?+4+ a’ )+a*=0, 
an equation which is symmetrical with respect to # and a’: 
hence, if we have 


24 
BA— CU= oa’, 
a 
we also have 
2 
BU CAs “- ae’. 


253. Curves of Double Curvature. A curve of 
double curvature is one, three of the consecutive elements 
of which do not lie in the same plane. Such a curve must 
be referred to three co-ordinate axes, and requires for its 
expression two equations which represent the projections of 
the curve on two of the co-ordinate planes. 

Let the equations of the curve be 


y=f(x) (1), %=9(%) (2); 
(1) being the equation of the projection on the plane (a, y), 
and (2) the equation of the projection on the plane: (@,/2)7 alt 
x, y, %, are the co-ordinates of a point of the curve, and 


CURVES OF DOUBLE CURVATURE. 435. 


e+Ax,y+Aay, z+ 2, the co-ordinates of an adjacent point, 
then, by the principles of solid geometry, the length of the 
‘ chord connecting these points is 


{ (amy? + (ay)? + (ae)? 
Then, if s is the length of an arc of the curve estimated from 
a fixed point up to the point (a, y, 2), that of the arc from the 
same fixed point up to the point (7+ aa, y+ Ay, z+ 42) 
will be expressed by s+As. We shall assume 


Jit a bee IER el 
(aa)? -+ (ay)? + (az)? 
Ag 
a= lim. eure Ove N 7 Te 1, 


fe ea 
a Lean es 


emp fa (2) + (2) ta 


The two equations of the curve enable us to express 


and therefore 


dy dz 
dx’ da’ 
in terms of «; and, by integrating, s will then be known in 
terms of «. 

Any one of the three variables may be taken as independ- 
ent; and the above formula may be changed into 


ff < (i) +) 
= nay 


436 INTEGRAL CALCULUS. 


When 2, y, and z are each a known function of an auxiliary 
variable, ¢, as may be the case, then 
dy dz 
dy __ Ke at dz __ at 
dx dz dx dx 
dt dt 


and we may have 


J}3+ Gz) +(@) }" 
oe) +) 
« aff) +) +G) pa 


254. To convert the formule of the last article into polar 


s 


| 


formule, take the pole at the origin of co-ordinates, and denote 
by 6 the angle that the radius vector makes with the axis of 
z,and by g the angle that its projection on the plane (a, y) 
makes with the axis of w; then we have the relations 


=r sin. cos. g,. ¥ =T'sin. 0 Sin. @, 2 =F iCosee 

These three equations, together with the two equations of 

the curve, make five between which we may conceive r and 

to be eliminated, leaving three equations between 2, y, 2, and 

6: hence, x, y, and z may be regarded as known functions of 6. 
Therefore 

dr : Hae 
7g BID 6 cos. p — or — sin. 6 sin. p 7 + 1 cos. 8 Cos. g, 
= sin. O sin. a + 7 sin.6 Cos. 9 - + rcos. sin. g, 


Z dr ; 
—=—cos.9— — 7 s1n. @: 


1 ON do 


POLAR FORMULA. AZ, 


aCe) 


dé 


which, by changing the independent variable, may become 


s=fin( 3) tite mela) 5 dr, 
or c= fir (GZ) + +(Z) 49 cnt dp 


255. Polar Formula for Plane Areas. In the 
curve BM of which the polar 
equation is r= (0), let r, 0, be 
the co-ordinates of the point WM, and 
denote by A the area bounded by 
the curve, the radius vector PB 
drawn to the fixed point B,and the F 
radius vector PM. Then (Art. 165) 

ak : =\9 (a) } eee A=, [{o(a) }'a9 

Let w(6) be ae function having (6) for its differential 
co-efficient; then 4=w(6)-+ (: and if A,, A,, denote the 
areas corresponding to the values 0,, 0,, of the vectorial angle, 
we have . 


da\2) 4 0 N C m™  % 
+r sino (TF) t dé ; 


A, = (9) +C, Ay= (2) + C; 
te A A= ¥(%)— 9H) =a fo fo} 


438 INTEGRAL CALCULUS. 


EXAMPLE 1. For the parabola, when the 
pole is at the focus, and the variable angle, 
measured from the axis, begins at the ver- 


9 tex, we have x=p—rcos.6, y=rsin.0; 


R 
from which, and the equation y? = 2pm of 
the curve, we get 
ners. L/L ps a ee ae ae 
esl c08 a 2s ae ot andes prin” 
"2 ng 
p? 6 0 2. gl pean 
— F f(2 + tan.? 5) sec. 5 Cope a tan. 5 = 79 18D 5 + C; 
p? 0, 6 2 6 6 
. 4,— A, == (tan 7 — tan. yl a 5 (tam ener tm), 
: 7 we We p? 
Making 6,=—0, 6, mage have for the area 7 + 172 3 
8 
Ex. 2. The equation of the logarithmic spiral being r = bee» 
we find 
1 *0 Dee 
—=— | bec SS Bie 
A 5 it ec do 7? + C, 


le. 20g! *he Mae 

jan} fined e 2 te 
256. A polar 
formula involving 
double integration 
may also be con- 
structed for plane 
areas. Suppose the 
area included be- 


tween the curves 


BME, bme, and the 


POLAR FORMULZ.. 439 


radii vectores PB, PH, is required. Divide the area up into 
curvilinear quadrilaterals by drawing a series of radii vectores, 
and describing a series of circles with the pole as a centre. 
Let no be one of these quadrilaterals, and denote the co-ordi- 
nates of n by 7, 6; and of o by r+Ar,6+ 6. Now, the area 
no is the difference between two circular sectors; and the 


: pe cee a. : 1 
accurate expression for this difference is rArad + 5 (Ar)? Ad, 


the ratio of the second term of which to the first is 
: (Ar)?a0 Ae 

rArdad 2r 
This ratio diminishes as Av diminishes, and vanishes when 
Ar =(: therefore we may take raraé as the expression for 
the elementary area, since, in comparison with it, the neg- 


lected term (ar) AO ultimately vanishes. 


_ 257. In the last article, it was shown that raraé might be 
taken as the expression for the polar element of a plane area. 
If we suppose this area to be the section of a solid by the 
plane (x, y), the column perpendicular to this plane, standing 
on the element raraé as a base, may be regarded as an ele- 
ment of the solid. The volume of this column is measured 
by zrarad; and therefore, for the volume V of the solid. we 


have V= f fzrdrdo. 
The value of z asa function of rand 0 will be given by 
the equation of the surface bounding the solid. 
EixXaMPLE. Required the measure of the volume bounded 
by the plane (a, y), and the surfaces having 
e?t+y?—az=0 (1), v?+y?—2x=0 (2), 
for their respective equations. 


440 INTEGRAL CALCULUS. 


Denoting the polar co-ordinates of a point in the plane 
(x,y) by rand 6, 0 being measured from the axis of 2, we 
have 


= '7cos:8, y =rsin.6 (3) ; 


therefore x? + y? = r*, which, combined with (1), gives 


From (2) and (8) we find r= 26cos.@: hence, for this exam- 


ple, we have 
| on = J feraran =z drdo. 


To embrace the entire volume comprised between the sur- 
faces indicated, the integral must first be taken, with respect 
to r, between the limits r = 0, r = 2bcos.6, since 6 is assumed 
as the independent variable; and then the integral of the 


result must be taken between the limits =F (= —5: 
Thus 

2bcos. 8 ; “ 

4 
V= oh drdé = 40 cos.4dd 0 
a a 
0 Bi ies A 
2 
4 4 
pas cos ‘9a9 =° lies (Art. 221). 
a 2a 


258. Suppose the polar element va7raé of a plane area to 
revolve through the angle 27, around the fixed line from 
which the angle 6 is estimated. A solid ring will thus be 
generated, the measure of which is 2zrsin.6raraAd; since, in 


this revolution, the point whose polar co-ordinates are 7, 0, 


POLAR FORMULZ. 44] 


will describe a circumference having rsin.@ for its radius. 
Denote by the angle which the plane of the generating 
element in any position makes with its initial position; then 
g + Ag will be the angle which the element in its consecutive 
position makes with the initial plane. That part of the whole 
solid ring which is included between the generating element 
in these two positions is measured by 


(po + Ag) r’sin. 6ArAd — pr* sin. Ora = r? sin. OArADAG. 


This may be assumed as the expression, in terms of polar 
co-ordinates, for an element of the solid: hence, for the vol- 


ume V of the whole solid, we have 


Kane falblan: Shee 


- in which the limits of integration must be so determined from 
imposed conditions, that the integral may embrace the entire 
solid to be found. 

Example. Required the volume of a tri-rectangular pyra- 
mid inasphere. Integrating the above formula, with respect 
to r, between the limits r=0,r—=a,a being the radius of 


the sphere, we find 


Beaty [fr sin. odrdoda =/fr sin. dédq. 


Now, a’ sin.@addqg is an element of the spherical surface; 
3 


and 5 sin.dA0Aq is therefore the expression for an elementary 
spherical pyramid having a” sin.@adaqg for its base. By this 
first integration, therefore, the element of the volume has 
changed from an element of the solid ring, generated by the 


revolution of rara§, to an elementary spherical pyramid. 
56 


442 INTEGRAL CALCULUS. 


Integrating next, with respect to 0, between the limits 


nm 
P=), 6= 5 
we have 
a? 
Vie = dq ; 
since 


0 
fsin.sd9 = — 008.0: Sf, sin, Gob ee 


By this second integration, the elementary volume has 


become a semi-ungula, or a spherical pyramid, having a bi- 
rectangular triangle for its base; the vertical angle of the 


triangle being Ag. 


We finally integrate, with respect to g, from — Oto ee 5 


and get for our result 
ma 


V=—3° 


SECTION IX. 


DIFFERENTIATION AND INTEGRATION UNDER THE SIGN /.— EULE- 
RIAN INTEGRALS. — DETERMINATION OF DEFINITE INTEGRALS 
BY DIFFERENTIATION, AND BY INTEGRATION UNDER THE SIGN /. 


(259. Wuarever function of x, f(x) may be, there exists 
another function, g(x), of «, such that g’(x)=—/(a);.and 


therefore | S@)dx = oa) + (7 (Art. 191), C being an arbi- 


trary constant. 
Denoting by wu the integral of f(x)dx, taken between the 


limits a and b, we have 


w= [fade = 9) — 9a). 


The definite integral w is independent of x, but is a func- 
tion of the limits a and 0; and its differential co-efficient with 
respect to either of these limits may be obtained without 
effecting the integration. For, since 

u= g(2) — 9(@), 
we have 
du ; du ; 
Ap gp’ (a); db gp’ (0) ; 
and, because g/(x) = /(zx), 


d 
ee 10): 


HOF du = f(b)db — f(a) da. 


443 


444 INTEGRAL CALCULUS. 


ILLUSTRATION. 
Let y=/(x) be the equation of the curve WN referred to 
the rectangular axes Ox, Oy. Ifa 
and 6 are the abscisse of the points 
Mand N, wu =e J (x)dx will repre- 
sent the area AMNB. Give to a and 


6 the increments 
AA' = Aa; BBGeEaoe 


Au AMM’ A’ au BNN'B 
AGcn aed Ae ee rae et 


The definite area AMNB is ‘obviously a decreasing func- 
tion of the first limit a, and an increasing function of the 


second limit 0: therefore 


Au... AMM'A' _ 


lim. < itn Tee ies — f(a), 
VA ee ea ine 
lim. Kia lim. PBR wert 0): 


Regarding the areas 4U/M/'A’, BNN'B’, as elementary, we 
see that the total increment of the area AMNB is the differ- 
ence of the increments that it receives at the limits. 


260. Suppose /(x) to contain a quantity, ¢, independent of 
»b 
x, and that the differential co-efficient of J J(x)dxe with re- 
spect tov is reauired. Replacing f(a) by /(a, t), we have 


U = cc, 0) On. 


If the limits a and 0 are independent of ¢ we have by giv- 


DIFFERENTIATION UNDER THE SIGN /f. 445 


ing to ¢ the increment Aé¢ (Au being the corresponding incre- 
ment of w), 


Au =]p f(a, t+ at)dx— f° f(a, t)dex 


sels (F(@, #442) — f(x, t)) da: 
A | AU maf J (a, satel! — f(x, Je 


e e 


Now, by Art. ay we may write 
J(%, t+ At) —f(&; t) ae t) 


At 


wie 


in which y is a quantity that vanishes we At vanishes. De- 
noting by 7’ the greatest of the values of 7, we have, gener- 
ally, 
7 
fide OS ays 

and, when neither a nor 0 is infinite, (b — a)7’, and therefore 

6 
{| ydx, will ultimately vanish: 

a 


est, (db. 2phdf (a, t) 
lim. = GH. i da. 


APPLICATION. 


Resuming the formula 


du _ dt _ pr df (x,t) 

po al eK gy  & ) 
just established, suppose g(z,¢) to be the function of which 
J (x, t) is the differential co-efficient with respect to x, and 
w(x,t) to be the function of which ae t) is the differential 


co-efficient with respect to x; then (1) becomes 


“ee AO = (b,t)—w(a,t) (2). 


446 INTEGRAL CALCULUS. 


If f(x, ¢) and a are both independent of b, (2) may be written 


se “ait @ fie w(b, t) (3) ; 


C denoting the sum of the terms which are independent of 6. 
Since we may give to 0 in (3) any value we please, replace b 
by «; then (3) becomes 


vio) = 9G) Ve (a). 


Dropping the constant C, which may be restored when neces- 
sary, and putting for the other terms of (4) their equivalents, 


[ease t) ets © fe, t) dx. 


we have 


da 


Example. Let /(x,t) = tha : 


d 
Ta? then J (, t)daxc — 


Jf, t) dx =e Fe = tan. hte 
d _ afl, 1, \_ paY(%,t) 
and 5, J A(t) de = at tan. tn) = f F da 


d 1 Qt? 
Valtere) = —Secpeey® 


Thus, having the value of (2s we find, by differentia- 


Te Q? 


tion, that of the more complex integral A paleait « ‘gir i —_—.— dx. 

261. When, in the integral vu = f f(a, t)dx, both a and b 
are functions of ¢, then zh will consist of three terms; since 
in this case, to obtain the total differential of w, we must dif- 
ferentiate it with respect to ¢, and also with respect to both 


DIFFERENTIATION UNDER THE SIGN /. 447 


a and b regarded as functions of ¢, and take the sum of the 
results. Thus we should have 


du gl t) du db , du da 
=! ct hoes 


df (x 
a " He, Per, te ce = f(a, t) eo (Art. 259). 


Under the above suppositions, the second and higher differ- 
ential co-efficients of w with respect to ¢ may be found. Thus, 
_ by differentiating each of the terms of the last formula with 
respect to t, we get 


Gu te) 2 


diay oe 
ero, re =A Wart se t) (a) 49 Te, t) i 
10 — a let _ 9 Vlad 
ILLUSTRATION. 


Let y = f(x, #) be the equation of the curve OD referred to 
the rectangular axes Ox, Oy, and 
y = f(a,t-+ at) that of the curve % x 
EF. Put 
One a, ON—'b, 
Meee Ag, NN‘ = Ab. 
Then w =p J (x, t)dx denotes 


the area MNDOC, and u + au the ° 
area M'N'FE: 
au = EE'F'F + DNN'F'— MM'E'C, 
Au EEE’ F  DNN'F MM EC. 


Abba aad At At 


448 INTEGRAL CALCULUS. 


It is plain that the first term in this value of is the ratio 


of Aé to the increment of the area due to the change from 
the curve CD to the curve HF. The limit of this ratio is the 


att of ip ACAI wa 1%) ge So, ‘alec the inn tenes 


second term is the limit of /(0, ¢) a and the limit of the third 


term is the limit of /(a, i) = —: hence 


du Hae t) da 
sf IAS dex 2+ 7 ,t) a ee 


which agrees with first formula established in this article. 

262. An indefinite integral may also be differentiated with 
respect to a variable contained in the function under the sign 
of integration which is independent of the variable to which 
the integration refers. 

Let the integral be u =f /(a, t)da, t being independent of 
a: then, without impairing the generality of this integral, we. 
may write 


u= fo f(a, dde+y); 


w(t) being an arbitrary function of 7. Differentiating with 
respect to ¢, ¢ not Lae on a, we have (Art. 260) 


a a= As 2D de + w(t): 


but, since w/(¢) is a constant with respect to a, it may be 


included in the constant of the integral ii a dx; and 


hence the last equation may be written 


of ZO aa, 


INTEGRATION UNDER THE SIGN ff. 449 


and we have only to differentiate the function under the sign if 


with respect to ¢. 
263. Integration under the Sign of Integra- 


b 
tion. Taking the definite integral J J(x,y)dx as the differ- 
ential co-efficient of y, and integrating, we have 
b 
favs. f(%, y) de 


for our result; and it is proposed to prove that this result is 
the same in whichever order with respect to x and y the inte- 


grations are performed; that is, we shall have 
b b 
fay f Seyde= J de f f(a, y)dy. 
df f “ y)dy 


For aJ. dic f f(a, y)dy = f de 
eel 


Integrating the two members of this equation with respect to 
y, we get 


Jere ydy = fay f f(x, y) dx 


and, if the limiting values of y are c and d, we shall have 
b d 
f wf Sandy 
ad b 
=f dy f fa, y) da. 


The figure gives the geometri- 
cal interpretation of this formula. 


Kither member represents the vol- 

ume AC’ included between the 

plane (x,y), the surface A’B'C'D' having z =/(a, y) for its 
57 


450 INTEGRAL CALCULUS. 


equation, and the planes whose equations arex=a, «= 68, 
y=e, y=". 

ExaMPLE 1. Find the form of the function g(#) such that 
the area included between the curve y= q(@), the axis of a, 
and the ordinates y = 0, y = g(a), shall bear a constant ratio, 
n, to the rectangle contained by the latter ordinate and the 
corresponding abscissa. 

By the conditions, we must have 


: __ ag(a) , 
Yio 2) to ag 


and, since this is to hold for all values of a, we may differen- 
tiate with respect to a: hence 


_ g(a) g(a) | 

De) a eee 
: PU) oa hey 
ne g(@) a? 


and by integration 
lp(a) =(n—1)la+C. 
Passing from logarithms to numbers, 
p(a) = Car: +, g(a) = Car; 
and the equation of the curve is y= Cau"—!. 


Ex. 2. Determine such a form for g(a) that the integral 


« g(x)dx ; : 
shall be independent of a. 
0 A (4—2) 5 


Put «= az; then, since the limits « = 0, x =a, correspond 


| Realee peso Ua] mead 


cae a g(ajdxe — prJ/ag(az)dz 
=/ iv@=o- J, waieam 


INTEGRATION UNDER THE SIGN /. 451 


By condition, wis to be independent of a: therefore the dif- 
ferential co-efficient of w, with respect to a, must be zero. 
But 


gige) / 
Ree 2a tN 8) yee Dy 
da a/(1 — 2) 2A4/(a — x) 

0 
and, since this last integral is to be zero for all values of a, 
we must have 


cp! 1 
p(x) + 2ap’(x)=0: .°. ae Horne 
Therefore l(a) = — ae + 0, 
or g(x) = . 


Let AOB be acycloid, with its vertex downwards; and let it 
be referred to the axis 
Ox, and the tangent 
through its vertex, as 


co-ordinate axes. Px 


Then, denoting the 
angle DOP by 6, we ¥ F aco 
have for the co-ordinates OF = y, OQ == of the point P, 
gee — Tp Go Cos 6 
yY—OF = AR —AK= AR — AD + ED 
= an — as + asin. 6. 
Put 6 =x — q, then these values of z and y become 
x—=a—acos.g (1), y=ap+asin.g (2). 
From (1) we find 


a 
Pi COS.1 


—x“ ., if 
sing = 7 | 2ax— ats 


452 INTEGRAL CALCULUS. 
and thus (2) becomes 
a—aX 
Y COS Se ear eae — 


which is the equation of the cycloid. By differentiation, we 


get 
dy Ree — we ae J dy\? oa 
a te ee ee ee fe 
da a da eh @ a’ 
and, by integration, s=+/8ax. We therefore conclude that 


g(x), in Ex. 2, is the expression for the arc of a cycloid esti- 
mated from the vertex. 


This example is the solution of the problem in mechanics 
for finding the curve down which bodies, starting from dif 
ferent points, will fall in equal times. 


264. The Eulerian Integral of the First Species 
is an integral of the form 


1 
i] aP—l(1 — x)2—" da, 
0 


in which p and q are positive numbers. This is denoted by 


B(p,; q). 
The Eulerian Integral of the Second Species is 
of the form 


co 
ii Mie be 2 5 
0 


and is denoted by I(n). 


The first species may be put under the two forms 


oo p—ld 5 @ 
i) aC 2 sin.??—! @ cos.47—-1 dp, 


by making «= i zi ioe the first form, and x = sin.”6 for the 


second. 


EULERIAN INTEGRALS. 453 


The integral of the first species is a symmetrical function 
of » and q; for, making «= 1 — y, we have 
0 
Bip, Q=f (Loy yt dy = Bg, p): 
B(p, 9) = BG, p)- 
265. Integrating by parts, we have 
for (1—a) de 
x? (1 — ax)! 


se 4 ~ q-1({ — 
=e sr ae) 1(1— x) de 


eee (1 — x)? PL fiee—3 (1 — ode —£. fer (1—2\t—I¢ 
= Z ca (1—2) x ad 2! x) L 


Therefore, taking 1 and 0 for the limits, we have 


AG a a q) = BP, q) 7B +h q): 


Bip +1, 9) = ma Bp, q). 
In like manner, 
B l= Se 
(pRI+l)= rae B( p,q): 


In the integral of the first species, therefore, each of the 
- exponents p and g may be diminished by unity. 
266. In the Eulerian integral of the second species 


oa) 
if (See sii Sa 9 hal 
0 


n must be positive, otherwise the integral would be infinite. 


For if n be negative, and equal to — p, we should have 


ie a) lo a} ye 
—2£ pmn—l pee é 
ip e-*a ot oa {aa : 


and it is plain, that, when w = o , the differential co-efficient is 
zero, and therefore the integral is zero; and, when # = 0, the 


454 INTEGRAL CALCULUS. 


differential co-efficient is infinite, and therefore the integral 
is infinite. 
The integral '(n-+1) may be made to depend on I(n). 
For, integrating by parts, we have 
fe-*atda = — e~*x" + nfe-*x"—" dx. 
But e-*a” reduces to zero both when « = 0 and when = 
(Ex. 3, Art. 103): therefore . 


ie) ie a) 
J Gap ra Lig ees nf Co * aaa 
0 0 


or r(n+1)=nI(n). 
In like manner, 
I(n)=(n—1)r(n—1), F(n—1) = (n— 2) T(r — 2); 


and, if 2 is entire, we shall have, finally, 
r(2)=r(1), r(1) ={), en? dpeeas 
0 F 


Therefore, when 7 is an entire and positive number, we shall 
have 

I'(n) =1.2.3...(n—1); 
and, if ” is a fraction greater than 1, then the formula 

I'(n) = (n—1)P(n—1) 
enables us to reduce the integral I'(v) to that of I'(u), u de- 
noting a number less than 1. Hence, to compute the value 
I(n), it is sufficient to know the values of this function for 
values of n between 0 and 1. 

267. By putting e~* = y, the integral I(r) may be made 

to take another form. Thus, from e~* = y, we get 


NAS nel hae cy we 1 yi sa 
flected f(y av= py dy 
n—Il 
or I'(n) =i (: 4 dy. 
0 


EULERIAN INTEGRALS. 455 


268. Relations between the two Eulerian Inte- 
grals. Assume the double integral 


{s 1h gPe+raq—t Dnt e—I+ne dyda, 


and integrate with respect to 2: it thus becomes 


riptaf a® ays (Art. 266). 


Integrating the same double integral with respect to y, it be- 


comes } 
© p—x p+q—-ld oo 
U(p) f= =r(p)f eat dx =I(p)I(q): 
therefore 
oe ; 
Peta), eT (Pyr (a): 
RU a an BC Pe a 
ia pie POD = pa 
that is, 


Pee tea rip icay 
|| Chass verter, 


Putting : for x in the first member of this last equation, 


we have 
eee Fe a) 
SC Penne aiy 
or if 2? -l(q — 2)I—dzg —= gP+¢—1 Pees 


269. The last formula in the preceding article is a particu- 
lar case of a more general formula by which may be expressed, 


in terms of I functions, the multiple integral 


Oe 2. Be (c —x—y—%...)* 'dadydaz... 
extended to all positive values of x, y, z..., which satisfy the 
condition « +y-+2...< a. 


456 INTEGRAL CALCULUS. 


Limiting ourselves to three variables, let 


a a—2x a—-x—-y 
A= p—\ df, I—ld, r—l(q—g — you Bie 
ip x of y yf, a"—*(a—a@ —y—a2)s dz 
Now, by the last article, 
ees = | = —y)rrs-t I'(r)I'(s) 


r—l wen easy eke s—1 —_ Mas bee: 4 
, e°—"(a—x—y—z)* =(Aa—zZ Tires 
Multiplying this by y?%~'dy, and integrating with respect to y 
from y = 0 to y= a—z@, the result is 
(a — w)etr+s—1 LAG) Dna 8 aya 
riqtr-+s) r(r+s) 


— (a ie io) ta eee 


(QI (r) Ts), 
(ace eee 
and finally, multiplying this last by «?—1da, and integrating 


with respect to « from x= 0 to x =a, we have 


— qgetatrr+s— LTipr(gnr(r)l(s) 
6 Te 


In this, making a= 1, s=1, we have 


p—1l,,q—lyr—l MAL I( p)U(q)r(r) 
Sf fe pits dati de Te dy 


the limits of integration being any positive values of a, y, z, 


which satisfy the inequality «+y+2< 1. 


a & y\P Z\7 
Assume " sai e Seay () =a fee 


Pha SLE eee 
then om 4=— Sf wer nb CY ductndt, 
rf 


subject to the condition that u on n+¢ <1: therefore 


gare HOGG) 


aby 1 ae (2). 
st ey 


EULERIAN INTEGRALS. 457 


270. By means of Formula 2 of the last article, we can 
find the volume bounded by the co-ordinate planes, and the 
surface having for its equation 


() +) +) =" 


PeeeenE When @—f—y7—2, and p=g=r=—1, the 
surface is that of an ellipsoid of which 2a, 2b, 2c, are the axes. 
_ Then, by the formula, the volume V of 4 of this ellipsoid 
will be 


But Ae a 1)= 7 =r e) =, ip () (Art. 266), 


and I" os a/2; for, let u al e—** dx, then also 
u => [\e7* dy, 
es 


and ur— [ eda [ e-” dy = wi (il @atacats dady. 

fer de fet ay= ff 

Now, flee-* —” dady is obviously } of the volume the 
04-9 


equation of whose surface is z=e-**—”*, In terms of polar 


co-ordinates, the expression for the same part of the volume is 


J fi ardodr = [ IWpen tie 
0 0 : : 


458 INTEGRAL CALCULUS. 


sas it 
and f d=6; 25) d= 48; 
» : 1 
us fe dt = 5 A/T. 
1 » 
Now, 7G} i) e—*x—*dx by definition, 
2 0 


==,9 [et dy = 20 = on by putting «= y?: 


__ abe | ryt __ abe 


therefore, Vis= 


8 Bye brea 
rst) 


271. Differentiation under the sign f enables us to find 


new integrals from known definite integrals. Thus, 


ay the nt 
EXAMPLE 1. il eee nn eee 
el 


Differentiating each member of this equation nm times with 


respect to a, we get 


74 HART tee gy () penta 3.9 2A 1 WA 
»@payt l= 5 a aati? 
2 da 1.3.5...(2n—1) a 
whence Cs i 3s 
i ("+ a)rrt 2.4.6...2n anes 
Bx. 2 freseee 
0 a 


After n—1 differentiations of the two members of this 
equation, it becomes 


ioe} 


f e~ * tle 2-3. ..(n — oe 
0 


thatis ft eon Cee (Art. 266). 


DIFFERENTIATION UNDER THE SIGN /. 459 


The last formula holds good when a is replaced by the 
imaginary quantity a+b —1, in which a is positive; for 
ee Ds dic 


C 

ares i 
__ e-* (cos.ba — / —1sin.b2) 
oe EL 7 aaa + C (Art. 13): 

therefore 
2 a 1 

—(a+bY—lex d. oe omer gs > Ls 

J, : aa + bf —1’ 


and, by differentiating this equality nm — 1 times with respect 
to a, we get 

1.2.3...(m — 1) 
(a+br/—1) ’ 

272. The formula just found leads to other integrals by 


ioe) 
ii e—(atoV—lz e®-\dx= 
0 


the separation of real from imaginary quantities. 
Assume a+b —1=p(cos.0+ / — 1sin.@), in which 


5 a ; b 
B=aV G?)- 52, cos.0 = Vato’ REY rs eae. 


Then 


z: : 
0 


= {Pa (cos.ba — s/ — I sin.ba) a*—! dx 
0 


and 
Wor, ; 10 —\ boy 27) Lee 
(atb/ —1)” p” cos.nd+/ —1sin.nd 
In) 


= —— (cos.nd — / —1sin.né): 
p 


460 INTEGRAL CALCULUS. 
.. if e~* (cos. ba — / — 1 sin. bx) 2" dax 
0 


= me (cos.nd — ny hess} sin. 70) ; 


an equation which may be separated into the two, 
P(n) 
pe 


eo 
i} Cy te SIN OL oe sin. 20, 
0 


n 


eS I'(na 
i e—* o"—! cos. bada = (7) cos.n0. 
0 Pp 


273. Making n=1 in the last formula of the preceding 


article, it becomes 


a —ax — ‘i e 
iP e€ cos. bada = ee 


therefore, denoting by c a constant less than a, we have 


if da Nee cos. bada =f se 


But 
Ip da je e—* cos. bada =|. diac ip e—* cos. bada 


@ Saeed = é ax , 
= i —_—_— cos. bada. 
0 4b 


Again: 


a ada: 1, cde 
J @apr3 co? + b?? 


wo e—c@ __ p—ar 1 a* +b? 
ip ee COR OL T 5 ee 


Making b = 0 in the last equation, it becomes 


2p 62) ea a 
(see 
iC 


0 w 


DIFFERENTIATION UNDER THE SIGN /. 461 


a result that may also be obtained by multiplying both mem- 
bers of the equation 


(s oe de = 


by da, and integrating the result between the limits a and ce. 
274. In like manner, from the formula 


2 ene Woe 1D 
f, e—“ sin. badx = at EB? 


we get 


f fF e-* sin.brde = fier 


a Cc 
— tan. — tam! —° 
b b 
But 
a wo A foo) a Fs 
|e f e~ sin. bada aah da { sin. bxe—“* da 
c 0 0 c 
2 e-—ce __ p—ar 
=) ee SIT OLS 
0 ax 
aes if ee gin, pvae — tan, + ——,tan.!— 
0 x b b- 


In this formula, making a =o, c= 0, it reduces to 


= sin. bax 
f sin. b de —™ 
faa Beer 2 
n 
when fe: when See the second member becomes — 9° 


gin. Es 


from which it is seen that the integral if dx changes 


abruptly from 5 to 5 when 8, in passing through zero, 


changes from positive to negative. 


462 INTEGRAL CALCULUS. 


275. The integral f e-# de = svn (Ex. Art. 270) leads 
tof e-"da=ar/n; for 


ihe e—™ da ape e-* dx +f. e—* da. 


Now, if we change x into — a, we have 
0 0 
J eda =f edz = ha/n: 
— 00 0 
i Sth tesla VE rg 


And generally, if f(x) is a function of the even powers of a, 
that is, such a function that /(x) = /(— «), then 


fo f@)de =2 [" f(a)dz; 
for f s@ae=f a flo)de +f" f(a)de. 
But fo fede =f" s(—2)de=f fe)de: 
f fla)ae = af f(a)de. 


In like manner, it may be shown that if f(a) is a function 
of the odd powers of a, that is, if f(— «) = —/(a), we should 
have 


f° S@)de= 0. 


276. In the integral [ e—" dx =/n, putting x/a for 


x, we have 


which, by n differentiations with respect to a, becomes 


ie eae" nn Jn oe Sip aN — 1) a (ake). 


DEFINITE INTEGRALS. 463 


In this, making a = 1, we have 


ii o-# ihdan = a/m 123:5--.(2n = 1), 


¢ 
on 


277. Changing x into x +a in the formula 
ecto — a/ tt 
of the preceding article, we get 
if Ga op afar 


ioe) 
e 2 2 
that is, G8 f Ct I AY Tee 
—o2 
at 2 d. 2 
nee e777 —2axr Tp = e*? 1. 
{jee x4 
o 2 0 7 ra) P 
But {i om Sat =f Cae Oe +f Camis 6 O07, 
ee mie 0 


0 ) ze 2 
and if er? da =}/ et +2ar da 
aap 0 


by changing x into — a: 
ai in AB OLN wens tr eo limcre ede +f eM de 
=o 0 : 0 


=P g-* (e%4* 1 6-392) do « 
0 
whence 
i e-=" (ere oo Crs) Oe “= ew Jt. 
J 0 
In this equation, replace a by a4/— 1; then, since 


ere | g— tar — e—WaV—1 | e2ae~—1 — 2 cos. Zan (Art. 73), 


we have 


cs 2 1 °o 
iP 6 * cosecccds — ae eae / 1. 
fd 


464 INTEGRAL CALCULUS. 


This example is another instance in which the value of a 
definite integral is found by passing from real to imaginary 


quantities. 
#78. Another process by which f e>#* cos. 2axdz may be 
9 * | 


found consists in differentiating with respect to @ and subse- 


quent integration: thus, put 


io 2) 
U =f e—* Cos. 20005 5 
0 


then 
ae = —f sin. 2axe—* 2ada =| sin. 2am, d,e—*. 


Integrating by parts, and observing, that, at the limits, 


° 2 
sin. 2@ce—* is zero, we have 


du v4 2 
iy eee ee —& 2actada = — : 
la J, cos. 2ax2ada 2au 
du 
WO, 99 
U 


But, regarding w as a function of a, we have 


du : 
ad, do. thing Wee 
Integrating with respect to a, we get 
lu=—a@?+C: 2. wae @ +e = Oe-e 


by making e*-= C. To determine C, make a= 0; then 


ms — x? — ae 1 — e 
wet ae dx = s/t = C3 
therefore 


le e—** cos, 2aada = ; e—% n/m. 
0 


SECTION X. 


ELLIPTIC FUNCTIONS. 


279. Elliptic Functions or Elliptic Integrals is 
the name given to the following integrals : — 


9 dé 
First order. mommll cll Koper A 
Ip /1 —c?sin.Z26 — ie ) 
9 ee 
Second order. if /1—c?sin26 d) = E(c, 6). 
0 ‘ 


Third order i : a9 
'¥ 0(1+asin20)/1—c?sin26 


The constant c is called the modulus of the function, and is 


SEIU, AE 6). 


supposed less than unity; the constant a, which appears in the 
third function, is called the parameter; and the variable 0 is 
called the amplitude of the function. The function is said to 


be complete when the limits of the amplitude are 0 and - 


The integral of the second order expresses the length of the 
arc of an ellipse estimated from the vertex of the conjugate 
axis (Art. 240); the semi-transverse axis being unity, and the 
eccentricity of the ellipse the modulus of the integral. From 
this fact, and from the relations which exist between the sev- 
eral functions, the term elliptic functions has been derived. 
Our limits permit us to investigate but a few propositions 
relating to such functions. 

280. Putting «x for sin.@, the integral of the first order 


becomes if. dx 
0/1 —eV/1—cx 


59 465 


466 INTEGRAL CALCULUS. 


In like manner, for another value of « denoted by ,, we 
have 


iP dx, 
Vil oo V1 — ctx? ot 
Now assume the relation 
da On 
Soe a ae iT athbaeur a . L0) ee 2 
V1 — a? / 1 — cx? V1 eV 1 ee 
Multiply through by the product of the ionotaeean divide 


=0 (1). 


by 1 —c?a’* ,» and integrate ; then 


Ne Walaa 7, | 
, omy a saa /1—c?a 


c da 
ia oata? P= cya 
= constant. 
Integrating the first term by parts, we get 
[PSR SG aoa j t/1— ai V/1—c?x 
- O — : 5) 
1—c*x*ax, 1—c?’x*x, 
+ foe, (+e*)(1tc?x?x! was —2c*x; ams 
At eee /1—2ir/1 —c? 2? 


es few eat ae V1 — 22/1 — e822? de. 

In this result, interchanging w and x,, we have the second 

term. Adding results, observing that by (1) the terms of the 
sum which are under the sign df reduce to zero, we find 

eV1—¢8 Vie +0718 V1—oa 


Z 
der. al als 


aa const. (2). 


Kq. 1 expresses the condition that the variables a and a, are 
so related that the sum of the integrals 


| a 
VI —eVi— V1 — ai Lene 


shall be constant. 


ELLIPTIC FUNCTIONS. 467 


“é dx 
Put if! Rae a i tetas — a, 4 os S(a), 


V1 —«?= O(a), V1— cx? = R(a); 


# dc 
Wey SBVisag a? B= 5) 


ea = O(8), V1—c?a’ = R(8). 
Then, by Kq. 1, we have 
da+tds—0: 
a + p= constant = ». 
It is also seen from (1) that the constant 7 is the value of 


also 


x, when «=0; and, further, when « = 0, we have 
e—0, B=7, 2, =S(7)—S(a+ 8): 
therefore, by making the proper substitutions in (2), it be- 
comes 
, S( ae) (8) B(6) Se Ta ) 
S(a + 8) = ce 
1 — o?{ S(a){* | S(B 


which is the fundamental formula as given i Euler in the 


theory of elliptic functions. 
281. Suppose the variables 6, 6,, to be connected by the 


equation 


is d9 4 os dd, Kod oa fil 
0/1 — c?sin.26 0 /1 —c?sin. 120) 04/1 — ec? sin2 0 
(1), 
or Ec, 6) + F(¢, 6;) = F(c¢, w), 
in which » isaconstant. If 6, 6,, be regarded as functions of 
a third variable ¢, and (1) be differentiated with respect to the 
latter variable, we have 
ad) dd, 
dt dt 
V/1 — c2sin20 5 /1 —c? sin, 


=0 (2). 


468 INTEGRAL CALCULUS. 


Since the new variable ¢ is arbitrary, let us assume 


dp = /(1 —c’sin26) (3); 


Obie 
whence, from (2), 
Oe 1 oss mee 
dt 
Squaring (3) and (4), and differentiating, we get 
i hes (Joa wv chy a 
Pah ye sin.dcos.9, i ee sin.f,Cos.4,: 
i Nek a has bad, 
*. on ae a, = — c*(sin.6 cos.6 + sin.6, cos.4,), 
“ihe CG? tae : 
OF a5 (9@+6,)=-—- 5 (sin. 29 + sin.20,) (5). 


Put 6+ 6,= gq, and 6—6,=w; then 
2= p+, B,=g—-Y, | 
sin. 20 = sin. 9 cos. w + Cos. p sin. w, 
sin. 24, = sin. g Cos. wy — COS. M SiN. w: 


therefore, from (5), we have 


9 


Ci GME IG an cos ee c” sin. w cos 
tae -@Q ~W, Wiiaae -W -Q. 


We also have 


dp dy _ au) ~(q) =2 1 —cos.29, _1—cos. 26 
dt ada) i) = ) 2 2 ) 


a COB: ASE, __ cos. 26;\ ain ae 
=o (S ; ) = _psin. w: 
d* op d?w 
dt® aS 
= COL. —=CObs Ms 
dp dw ’ dp dw z 


dt dt dt dt 


ELLIPTIC FUNCTIONS. 469 


d?w 

d dg. d - dw. “at? 

But psn. g = copa, alae ar: 
dt 


ad /,d¢ d d /,dw GN 
ee yea pati 
a) = Roe aA? i) Shard 


Whence 


1? —Isin w+ C, ls 7 isin. p + Ch; 


or by putting C= LA, Cr—l4y, i passing from logarithms 


to numbers, 


fA BIN. tH, oF — A, sin. 9 (6): 


| . dy 
*, Asin. w Fiscme A, sin. Pa 


Acos.w= A,cos.g+C (7). 

From Eq. 1 wo see that F'(c,6) = F'(c,~) when 6,=0: 
therefore we then have 6 == gy =y, and (7) then becomes 
(4 — A,) cos. 4 = C; and therefore 

A cos. (6 — 6,) — A, cos. (6 + 6,) = (A — A,) cos. pn; 
whence, by developing cos.(#—6,), cos.(9@+4,), and re- 
ducing, : 

(A — A,) cos. 6 cos. 0, + (A + A)) sin. 6 sin. 4, 
= 4.— A) COs. 6 (8). 


Now, 
dy Oiaps. qed fo 
oa = ieee F = /(1 — c* sin.’ ¢) — 4/(1 — ec? sin.’ 6,), 
dw 


e, = (1 — ce’ sin.’ 6) 4+ (1 — c? * sin. rye 
Substitute these values in (6), and make 6, = 0; then 
/(1 — ec? sin.) — 1 = Asin. p, 
a/(1 — ec? sin.) + 1 = 4,sin. pw. 


470 INTEGRAL CALCULUS. 


From these equations, getting the values of d+ 4,,4— A,, 
and substituting in (8), we get, finally, 

cos. 6 cos. 6; — sin. @ sin. 6; 4/(1 — ec’ sin.) = cos. (9). 

This relation, by an easy transformation, may be made to 
take the form | 

cos. 6 = cos 9, cos. w + sin. 6; sin. wa/(1 — ec’ sin’) (10). 

Eqs. (9) and (10) express the connection which exists be- 
tween the variables in two elliptic functions of the first order 
which have a common modulus. 


282. Let Fc, 0), '(c, 9,), be two elliptic functions in which 
c, c,, and 6, 6,, are connected by the equations 


9 4c sin. 24, 
It is proposed to prove that 
2 
ANG Ey ote oe 


Differentiate Eq. 2, regarding 6, as the independent variable ; 


then 
1 dd _ 2(1 +-ccos. 24) 
cos.26 d9, (e+ cos. 26,)? — 


From (2) we also get 
(c+ cos. 29,)? 
1 + 2ccos.20,-+ 0? 


dp 2(1-+ccos. 26,) 
db,) di 2eicosee; 


cos.2 46 = 


Also, from the same equation, we get 
c* sin? Bas 
1 + 2c cos. 26, + ¢? 
__ 1+ 2c cos. 26, + ¢? cos.’ 29, 
‘ap 1 + 2ccos. 20, +c? 


1— ce’? sin? 6=1— 


ELLIPTIC FUNCTIONS. 471 


f da 
fas (1) — ¢? sin.’ 6) 
2(1 + ccos. 24,) / (1 + 2c cos. 20, + c?) 


a 1+ 2ccos.26,; +c? 1+ 6ccos.24, a9 
a/(1 + 2c cos. 26; + c”) 
anne dd, 
i pag 7 phe ote ae 
pia al pe ea sin.?6, 
Gls e): 
d Ac , 
But the last integral, when ito: =c_, becomes 
2 do, 9 
l+e /(1 — ¢; sin.?4,) 1+ec (C1 91) 


| 9 
E(c, ea eee ole e i: 


If we suppose 6, = > 
2 ae 4 Tt 
aera e115) = Fle =) =2F ° 5) 
283. Having shown (Art. 281) that there exists, between 
the variables of two elliptic functions of the first order having 


a common modulus, the relation 
cos. 6 cos. 0, — sin. sin. 0, 4/(1 — c? sin. w) =cos.u (1), 
then, between the corresponding functions of the second 
order, there exists the relation 
E(c, 6) + E(c,6,) — E(c, v) = ce? sin. 6 sin. 6; sin. 
From the equation between the amplitudes 6, 6,, 0,, may 


be considered as a function of 6; that is, we may assume 


Li(c, 6) oe Lic, a) = Li(¢, 4) = (4), 


472 INTEGRAL CALCULUS. 


and differentiate, thus getting 
i] 
/(1 — ce? sin.’ 6) + 4/(1 — ¢* sin.? 6,) a 7G 


By Kq. 10, Art. 281, the first member of this equation may 
be put under the form 


cos. 9 — cos. 6; cos. u , cos. 0, — cos. 6 cos. u dd, 


sin. 6; sin. u sin. 6 sin. pu do 


__ @(sin.’6 + sin.’ 6, + 2 cos. 6 cos. 6, Cos. u 1 
a do 2sin.@ sin.6, sin.w 


But putting Eq. 1 under the form 
cos. 7 cos. 6; — cos. u = 4/(1 — c? sin.? #) sin. 0 sin. 0,, 
and squaring, we get 
cos.”? + cos.79: + cos.2 4 — 2 cos. 6 cos. 4, cos. u 
= (1 —e?sin.? 2) sin.7 08m oy 

Adding cos.’ 6, cos.’ to both sides of this equation, transpos- 
ing, and reducing by the relation cos.? = 1 — sin.2, we find 
sin.’ 6 + sin.’ 0, + 2 cos. 6 cos. 0; Cos. u 

= 1 + cos.’ u +c? sin.? 6 sin? 6, sin? n, 


& (1 + cos.” u + c? sin.’ 6 sin.? 6, sin? 1) 


2 sin. 6 sin. 0, sin. u 
d(sin. @ sin. 6;) 
a9 : 


== On 


~1 .  d(sin. 6 sin. 0) 
pag f (6) =c? sin. u Wi is 4 


and therefore, by integration, 


—f (0) =e? sin. 6 sin. 0; sin. pw. 


ok 


¥ 
ve 
% 


be iee 
Peter. f. 
a4 


PAP + 


